Random enriched trees with applications to random graphs

We establish limit theorems that describe the asymptotic local and global geometric behaviour of random enriched trees considered up to symmetry. We apply these general results to random unlabelled weighted rooted graphs and uniform random unlabelled $k$-trees that are rooted at a $k$-clique of distinguishable vertices. For both models we establish a Gromov--Hausdorff scaling limit, a Benjamini--Schramm limit, and a local weak limit that describes the asymptotic shape near the fixed root.


Introduction and main results
The Brownian continuum random tree (CRT), which was introduced in the series [Ald91a, Ald91b,Ald93] of seminal papers by Aldous, is known to be the scaling limit of a large variety of combinatorial structures, see [HM12,Stu,PS,Car14,JS,Bet,CHK]. The study of these combinatorial objects is often done by using a suitable bijection in order to encode or approximate the class considered by simpler structures. Panagiotou, Stufler, and Weller [PSW14] showed that a large family of random graphs, so called random graphs from subcritical block-classes of graphs, admit the CRT as scaling limit. An important step in this project was the discovery, that encoding connected graphs as R-enriched trees and applying a decomposition due to Labelle [Lab81, Thm. A] allows for a particularly short and elegant proof of the main result. Motivated by this success, the present paper aims to study a general model of random R-enriched trees and metric spaces associated to these objects. We provide numerous applications that underline how our results provide a unified framework for the study of a great variety of combinatorial structures.
The concept of R-enriched trees was introduced by Labelle [Lab81] as a means to provide a combinatorial proof of the Lagrange inversion formula. Roughly speaking, given a class R of combinatorial objects, an R-enriched tree is a rooted unordered tree together with a function that assigns to each vertex an R-structure on its offspring set. For example, the structure can be a linear or cyclic order, a graph structure, or any other combinatorial construction. We may consider such a decorated tree as labelled, with a fixed finite set of labels, or up to symmetry. This is done in the same fashion as we may consider rooted unordered trees as labelled or up to isomorphism. If we assign a nonnegative weight to each R-structure, we may draw a labelled or unlabelled R-enriched tree of a given size randomly with probability proportional to the product of its weights. An R-enriched tree may be transformed into a random metric space by, roughly speaking, patching together independently drawn random metrics on its offspring sets which may depend on the R-structures.
Our first main result is that, under fairly general conditions, these models of metric spaces converge (after a suitably rescaling) weakly to the continuum random tree in the Gromov-Hausdorff sense. We provide sharp tail bounds for the diameter, which ensure convergence of extremal parameters not just in distribution, but also in higher moments. The second main result is that our model of a random labelled enriched tree converges locally to an infinite enriched tree, which we construct by a coupling with the limit object of simply generated trees. We apply our results to various combinatorial classes, in particular to random labelled and unlabelled graphs from block-classes, whose connection to simply generated trees is further elaborated.
The study of random labelled random graphs from block-classes has received some attention in the recent literature, see e.g. [MS14, PSW14, DN13, DFK + 11, DGN11, DGN + 12, PS11]. Our main application is to first passage percolation on random unlabelled rooted graphs from a block-class of graphs that is subcritical in the unlabelled case, a term which has been introduced by Drmota, Fusy, Kang, Kraus and Rué [DFK + 11, Ch. 5]. These classes of graphs include outerplanar graphs, cacti graphs and series-parallel graphs. In particular unlabelled random outerplanar graphs have also been studied by Bodirsky, Fusy, Kang and Vigerske [BFKV07]. While all previous results focus on enumeration and additive parameters, such as the degree sequence and the number of edges, blocks or cutvertices, we obtain a strong result on their global geometric shape: We show that random graphs from these classes decorated with independently drawn edge weights converge towards the continuum random tree and give sharp tail-bounds for their diameter. In particular, all higher moments of the rescaled diameter converge.
Moreover, we obtain a scaling limit for random Pólya trees drawn with probability proportional to the product of weights assigned to the vertex outdegrees. This covers the case of uniform Pólya trees with and without vertex outdegree restrictions, which has previously been established by Marckert and Miermont [MM11] for unordered binary trees, Haas and Miermont [HM12] for unrestricted and d-ary unordered trees and Pólya trees with outdegrees in a set of the form {0, 1, . . . d}, and, recently, also for arbitrary degree restrictions by Panagiotou and Stufler [PS]. Besides generalizing the main results of Panagiotou, Stufler and Weller [PSW14], the scaling limit of labelled enriched trees also applies to the loop-tree of a conditioned (on having n vertices) critical Galton-Watson tree whose offspring distribution has finite exponential moments. This result has previously been obtained by Curien, Haas and Kortchemski [CHK, Thm. 14] using different techniques.
Applying the local convergence of random enriched trees yields that random labelled graphs from block-classes converge locally to an infinite limit graph. If the graph class is subcritical, then this allows us to obtain the limit distribution of various parameters. For example, the limit distribution of the degree of a uniformly at random chosen root, which has been established previously by Bernasconi, Panagiotou and Steger [BPS09] using different methods. Furthermore, by building on results for simply generated trees due to Janson [Jan12] and Jonsson and Stefánsson [JS11], we obtain scaling limits for the size of the ith largest block of planar graphs, which seems to be a new result for the case i ≥ 2. The scaling limit for the size of the largest block was obtained by Giménez, Noy and Rué [GNR13] and the size of the second largest block was previously only known to be O p (n 2/3 ) and w.h.p. larger than n 2/3 / log(n), see [GNR13] and Panagiotou and Steger [PS10]. We also obtain results regarding the diameter of the block-tree of planar graphs. More precisely, we show that if h n denotes the maximum number of blocks through which a path starting at a uniformly at random chosen root may pass, then h n / log(n) converges in probability to a constant. The main ingredient in the proof is a result due to Kortchemski [Kor] who showed such a limit for the height of non-generic conditioned Galton-Watson trees. We also show that a conjecture of McDiarmid and Scott [MS14,p. 4] about the diameter of the block-tree of a random graph from a block-class would follow, if a question posed by Janson [Jan12,Problem 21.9.] about the height of simply generated trees could be answered in the affirmative.
1.1. Random weighted R-enriched trees. We state our results using the framework of combinatorial species introduced by Joyal [Joy81], which allows for a unified treatment of a large class of combinatorial objects and is particularly well-suited for treating symmetries. Formally, a combinatorial species is a functor F from the groupoid of finite sets and bijections to the category of finite sets and maps. Informally speaking, F is a rule that produces for each finite set U a finite set F[U ] and for each bijection σ : U → V a bijective map F[σ] : F[U ] → F[V ] such that the following properties hold. 1) F preserves identity maps, that is for any finite set U it holds that 2) F preserves composition of maps, i.e. for any bijections of finite sets σ : U → V and σ : V → W we require that While this framework surely seems rather abstract and complicated at first, it actually provides a very clean and powerful method of analysing the structure of combinatorial objects such as trees, graphs and planar maps, whose study from a modern probabilistic viewpoint has gained much attention in recent literature. We say a combinatorial species F maps any finite set U of labels to the finite set F[U ] of F-objects and any bijection σ : U → V to the transport function F[σ]. Given objects m U ∈ F[U ] and m V ∈ F[V ] with F[σ](m U ) = m V we say m U and m V are isomorphic and σ is an isomorphism between them. We say the object m U has size |m U | = |U |. An unlabelled F-object or isomorphism type is an isomorphism class of F-objects. For example, we may consider the species of finite simple graphs that maps any finite set U to the set of simple graphs with vertex set U . In this context, the size of a graph is its number of vertices. Any bijection of finite sets is mapped to the relabelling bijection between the corresponding sets of graphs. The labelled and unlabelled objects of this species are labelled and unlabelled graphs in the usual sense.
We are going to study random labelled F-objects over a fixed set or random unlabelled F-bojects of a fixed size, drawn with probability proportional to certain weights. To this end, we require the notion of a weighting of a species. Letting A = R ≥0 denote the nonnegative real numbers, an A-weighted species F ω consists of a species F and a weighting ω that produces for any finite set U a map for any bijection σ : U → V . Any object m ∈ F[U ] has weight ω U (m) and we may form the inventory ω U (m).
By abuse of notation we will often drop the index and write ω(m) instead of ω U (m). The weight of an unlabelled F-object is defined to be the weight of any representative. The inventory |F[n]| ω is defined as the sum of weights of all unlabelled F-objects of size n.
We are specifically interested in species having a tree-like decomposition, that is, we are going to study random R-enriched trees. Let R be a combinatorial species. We may form the species of R-enriched trees A R as follows. Given a finite set U let A R [U ] be the set of all pairs A R = (A, α) with A a rooted unordered tree with labels in U and α a function that assigns to each vertex v of A with offspring set M v an R-structure α(v) ∈ R[M v ]. For example, if R = SEQ is the species of linear orders then A R is the species of ordered rooted trees.
Let κ be a weighting on the species R. Then we obtain a weighting ω on the species A R given by For any n we use the notation [n] := {1, . . . , n}. If |A R [n]| ω > 0 we may consider the random labelled enriched tree A R n and the random unlabelled enriched treeÃ R n , drawn with probability proportional to its weight among all labelled objects from A R [n] or all unlabelled objects with size n, respectively. We give several examples, that fall under this model of random trees. a) (simply generated trees) Given a weight sequence w = (ω k ) k∈N 0 of nonnegative real numbers, we may consider the weighting κ on R = SEQ that assigns weight ω k to each linear order of size k. Then A R is the species of ordered rooted trees andÃ R n is the random simply generated plane tree T n with size n, see for example Janson [Jan12] for a survey on this model of random trees. Note that the result of discarding the labels of the tree A R n is distributed like the simply generated treeÃ R n . b) (Pólya trees) Let R = SET denote the species given by SET[U ] = {U } for any set U . Then A R is the species of rooted unordered trees. Again, we may consider a weight sequence w = (ω k ) k and assign a nonnegative weight ω k to any object of size k. ThenÃ R n is the random unordered unlabelled tree such that any Pólya tree A with n vertices gets drawn with probability proportional to v∈A ω d + A (v) with d + A (v) denoting the outdegree of a vertex v. Note that setting weights to zero allows us to impose arbitrary degree restrictions. c) (random graphs from block classes) A block of a graph is a maximal connected subgraph that does not contain a cutvertex of itself, i.e. deleting any vertex does not disconnect the block. Let C be a species of connected graphs with transport functions given by graph isomorphisms. We say C is block-stable or a block class of graphs if any connected graph belongs to C if any only if all its blocks do. Prominent examples are graph classes such as planar graphs, series-parallel graphs or outerplanar graphs. Let C • denote the species of rooted graphs from the block class C with transport functions given by graph isomorphisms that preserve the roots and let B denote the species of graphs from C that are either 2-connected or a single edge with its ends. The well-known decomposition Figure 1 allows us to identify the species C • with SET • B -enriched trees. See Section 2.4 for an explanation of notation and the involved operators. Details for this decomposition can be found in [HP73], [ We may define a metric d on the vertex vertex set V (A) that extends the metrics δ(v) by patching together as illustrated in Figure 2. Formally, this metric is defined as follows. Consider the graph G on V (A) obtained by connecting any two vertices x = y if and only if there is some vertex v of the tree A with x, y ∈ U v and assigning the weight δ(v)(x, y) to the edge. The resulting graph is connected and the distance of any two vertices a and b is defined by the minimum of all sums of edge-weights along paths joining a and b in the graph G.
We now introduce our model of random metric spaces associated to random enriched trees. Let R κ be a weighted species such that the weight sequence w = (ω k ) k given by ω k = |R[k]| κ /k! satisfies ω 0 > 0 and ω k > 0 for some k ≥ 2. Consider the weighting ω on the species A R introduced in Section 1.1, that is ω(A, α) = v∈[n] κ(α(v)) for all (A, α) ∈ A R [n]. For any integer n ≥ 0 with |A R [n]| ω > 0 we form the random R-enriched tree A R n = (A n , α n ) drawn from the set A R [n] with probability proportional to its ω-weight. Suppose that for each finite subset U ⊂ N and each R-structure R ∈ R[U ] we are given a random metric δ R on the set U ∪ { * U } with * U denoting an arbitrary fixed element not contained in U . For example, we could set * U := {U }. We may form the random metric space X n = ([n], d Xn ) as follows. For each vertex v of A n with offspring set M v let δ n (v) be the metric on the set M v ∪ {v} obtained by taking an independent copy of δ α(v) and identifying * Mv with v. Let d Xn denote the metric patched together from the family (δ n (v)) v as described in the preceding paragraph.
In order for this to be a sensible model of a random tree-like structure we require the following two assumptions.
(1) We assume that there is a real-valued random variable χ ≥ 0 such that for any Rstructure R the diameter of the metric δ R is stochastically bounded by the sum of |R| independent copies χ R 1 , . . . , χ R |R| of χ.
(2) For any bijection σ : U → V of finite subsets of N and for any R-structure R ∈ R [U ] we require that the metric δ R[σ](R) is identically distributed to the push-forward of the metric δ R by the bijectionσ : The first requirement ensures that the metric space X n maintains a tree-like structure and the second that the symmetries of the enriched tree do not influence the choice of the random metrics. LetÃ R n denote the random unlabelled R-enriched tree drawn with probability proportional to its ω-weight. We may form the random metric space Y n by choosing any representative ofÃ R n whose label set is a subset of N and forming the patched together metric d Yn as in the labelled case. By assumption (2) the distribution of the isometry class of the result does not depend on the choice of the representatives.
In order to formulate the following theorem we make use of the classification of weight sequences introduced by Janson [Jan12, Ch. 8] into three cases I-III with a subdivision of the first case into Ia and Ib or Iα and Iβ. See Section 2.1 for details. Roughly speaking, case Ia means that the simply generated tree corresponding to the weight sequence is equivalent to a critical Galton-Watson tree whose offspring distribution has finite exponential moments and the case Iα corresponds to a critical offspring distribution which only needs to have finite variance. Let span(w) denote the greatest common divisor of all integers i with ω i > 0.
Theorem 1.1. Suppose that the weight sequence w has type Ia. Then the rescaled space X n / √ n converges weakly to a constant multiple of the (Brownian) continuum random tree T e with respect to the Gromov-Hausdorff metric as n ≡ 1 mod span(w) tends to infinity.
The constant scaling factor is made explicit in the proof of Theorem 1.1 in Section 3.2. In order to ensure that extremal parameters of the rescaled (pointed) metric space such as the height and diameter converge not only in distribution, but also in higher moments, we show the following tail-bound.
Lemma 1.2. Suppose that the weight sequence w has type Iα. Then there are positive constants C and c such that for all n and x ≥ 0 it holds that P(D(X n ) ≥ x) ≤ C(exp(−cx 2 /n) + exp(−cx)). Note that if the random variable χ is bounded, then P(D(X n ) ≥ x) = 0 for all x larger than a constant multiple of n and hence it follows that there are constants C, c > 0 with P(D(X n ) ≥ x) ≤ C exp(−cx 2 /n) for all n and x ≥ 0. The requirements of Lemma 1.2 are slightly weaker than in Theorem 1.1, since we did not assume exponential moments.
In order to formulate the next theorem we need the concepts of ordinary generating series and cycle index sums, which we recall in Section 2.4 below. LetÃ ω R (z) denote the ordinary weighted generating series of A ω R and Z R κ (s 1 , s 2 , . . .) the weighted cycle index sum of R κ . IfÃ ω R (z) has radius of convergence ρ > 0 then Lemma 3.3 in Section 3.2.2 below implies that ρ < ∞ andÃ ω R (ρ) < ∞. The following results treat enriched trees considered up to symmetry. Theorem 1.3. Suppose that the ordinary generating series generating seriesÃ ω R (z) has radius of convergence ρ > 0 and that the series Then the rescaled space Y n / √ n converges weakly to a constant multiple of the (Brownian) continuum random tree T e with respect to the Gromov-Hausdorff metric as n ≡ 1 mod span(w) tends to infinity.
An explicit expression of the scaling constant Theorem 1.3 is given in the corresponding proof in Section 3.2. We also give the following sharp tail-bound for the diameter.
Lemma 1.4. Under the same assumptions of Theorem 1.3 there are constants C, c > 0 such that for all n and x ≥ 0 it holds that Again it holds that if χ is bounded, then we have the tail-bound for some constants C, c > 0. Our results have numerous applications, which we are going elaborate in the following subsection.
1.2.1. Applications. Let C denote a block-class of connected graphs and B its subclass of graphs that are two-connected or a single edge with its ends. The class of rooted graphs C • satisfies an isomorphism C • X · (SET • B )(C • ) and may thus be identified with R-enriched trees with R := SET • B . The correspondence is illustrated in Figure 1, see Section 1.1 for more details.
Suppose that we have a weighting γ on the class B, i.e. for each B-graph B we are given a weight γ(B) ≥ 0 such that the weights of isomorphic graphs agree. This induces a weighting κ on the species R by setting the weight of a set of graphs the product of the individual weights. Hence we have a weighting ω on C given by ω(C) = B γ B for all C-objects C, with the index B ranging over all blocks of the graph C. Let C ω n denote the random graph from the set C[n] drawn with probability proportional to its ω-weight. Moreover, let U ω n denote the random unlabelled rooted graph drawn from the unlabelled C • -objects of size n with probability proportional to its ω-weight. We apply our results to first-passage percolation on graphs. Let ι > 0 denote a random variable which has finite exponential moments. Given a connected multigraph G we may form the edge-weighted multigraph ι.G by assigning an independent copy of ι to each edge of G. The graph ι.G is again a metric space with the distance of any two vertices x, y given by the minimum of all sums of weights along paths joining x and y. Theorem 1.3 and Lemma 1.4 and the fact, that the diameter and height of the CRT are related by readily yield the following result.
Corollary 1.5 (First passage percolation unlabelled rooted random graphs). Suppose that the weighted ordinary generating seriesC •ω (z) is not a polynomial and has radius of convergence ρ > 0. Moreover, suppose that the series E(z, u) = zZ R κ (u,C •ω (z 2 ),C •ω (z 3 ), . . .) is finite at the point (C •ω (ρ) + , ρ + ) for some > 0. Then there exists a constant a > 0 such that in the Gromov-Hausdorff sense as n ≡ 1 mod span(w) becomes large. Furthermore, there are constants C, c > 0 with for all n and x ≥ 0. In particular, the rescaled height and diameter converge in the space L p for all p ≥ 1. We have asymptotically In particular, this applies to the uniform random graph from a graph class that is subcritical in the unlabelled case in the sense of [DFK + 11, Ch. 5]. Note that in the labelled case it does not matter, whether we draw random rooted or unrooted graphs with probability proportional to their weight, whereas in the unlabelled case these are different models of random metric spaces, in the same sense as Pólya trees differ from unlabelled unrooted trees. However, it is reasonable to expect that random unlabelled unrooted graphs also admit the CRT as scaling limit, with precisely the same scaling factor, as this was proven for the case of random unlabelled unrooted trees in recent work by the author [Stu]. Another application of Theorem 1.3 and Lemma 1.4 are "simply generated" Pólya trees, i.e. random Pólya trees drawn with probability proportional to the product of weights corresponding to the vertex outdegrees.
Corollary 1.6 ("simply generated" Pólya trees). Let (κ i ) i∈N be a sequence of nonnegative weights with κ 0 > 0 and κ i > 0 for some i ≥ 2. Hence κ can be seen as a weighting on the species SET. Let d denote the greatest common divisor of the set of all indices i with κ i > 0. For n ≡ 1 mod d large enough we may draw a random Pólya tree τ n having n vertices with probability P(τ n = τ ) proportional to v∈τ κ d + τ (v) for any unlabelled unordered tree τ with size n. If the power seriesÃ(z) given byÃ(z) = zZ SET κ (Ã(z),Ã(z 2 ), . . .) has radius of convergence ρ > 0 and satisfies Z SET κ (Ã(ρ) + ,Ã((ρ + ) 2 ),Ã((ρ + ) 3 ), . . .) < ∞ for some > 0, then there is a constant a > 0 such that with respect to the Gromov-Hausdorff metric as becomes large. Moreover, there are constants c, C > 0 such that for all x ≥ 0 and n it holds that P(D(τ n ) ≥ x) ≤ C exp(−cx 2 /n). Note that setting weights to zero allows us to impose arbitrary degree restrictions. The scaling limits of uniformly drawn Pólya trees with and without degree restrictions were established by Marckert and Miermont [MM11] for binary Pólya trees and by Haas and Miermont [HM12] for the unrestricted case and vertex-outdegrees in a set of the form {0, d} or {0, 1, . . . , d} for d ≥ 2. Later, Panagiotou and Stufler [PS] gave a proof that works for arbitrary degree restrictions. A similar tail-bound for their diameter was given by Stufler [Stu, Lem. 1.6]. Given a plane tree τ , the corresponding looptree Loop(τ ) is the multigraph obtained by deleting all edges and connecting any non-leaf with its offspring in a circular order [CK14a]. See Figure 3 for an illustration. The looptree associated to certain models of random trees is known to satisfy a scaling limit with the limit object given by a random stable looptree [CK14b]. We may study first passage percolation on a discrete looptree by considering the random edge-weighted graph ι.Loop(τ ). Theorem 1.1 and Lemma 1.2 readily imply the following result.
Corollary 1.7 (First passage percolation on looptrees). Let τ n denote a critical Galton-Watson tree conditioned on having n vertices, with the offspring distribution having finite exponential moments. Then ι.Loop(τ n ) converges weakly towards a constant multiple of the CRT. There are constants C, c > 0 such that for all x ≥ 0 and n we have the tail bound P(D(ι.Loop(τ n )) ≥ x) ≤ C exp(−cx 2 /n) and hence E[D(ι.Loop(τ n ))] ∼ a √ n for some positive constant a.
Curien, Haas and Kortchemski [CHK, Thm. 13] proved a scaling limit for Loop(τ n ) as a byproduct for their proof of the scaling limit of random dissections using different methods. The definition of the looptree considered there differs slightly from the above and may be obtained by, roughly speaking, removing the root of τ from the looptree Loop(τ ) and joining the lose ends of the adjacent edges. Hence, regarding the scaling limit, it does not matter which definition we use.
We also obtain the scaling for random graphs from block-classes in the labelled case. Corollary 1.8 below generalizes results from Panagiotou, Weller and Stufler [PSW14, Thm. 6.1 and 7.1] who studied the uniform case by also applying the concept of R-enriched trees.
Corollary 1.8 (First passage percolation on labelled random graphs). If the weight sequence w has type Ia, then the random graph with edge weightsĈ ω n converges towards a constant multiple bT e of the CRT in the Gromov-Hausdorff sense as n ≡ 1 mod span(w) becomes large. Moreover, there are constants c, We gave applications to various models of random trees and graphs. Our results may also be applied to certain subclasses of planar maps. A corresponding paper is currently in preparation.
1.3. Local Convergence. We are going to show that our model of labelled random enriched trees converges towards a limit object in a similar fashion as simply generated trees do. Throughout this section, let R κ be a weighted species such that the weight sequence w = (ω k ) k with ω k = |R[k]| κ /k! satisfies ω 0 > 0 and ω k > 0 for some k ≥ 2. Moreover, let ω be the corresponding weighting on the species A R of R-enriched trees.
In order to formalize convergence it is convenient to work with objects that we will call R-enriched plane trees in the following, i.e. pairs (T, β) of a plane tree T and a map β that maps each vertex v of T to an R-structure β(v) ∈ R[d + T (v)] with d + T (v) denoting the outdegree. Recall that any plane tree may be viewed as a subtree of the infinite Ulam-Harris tree U ∞ defined as follows. The vertex set is given by V ∞ = ∞ k=0 N k , the set of all finite strings i 1 , . . . , i k of positive integers. The empty string ∅ is the root and the ordered sequence of sons of any vertex v is given by v1, v2, . . .. Considering the plane tree T as a subtree of U ∞ allows us to interpret T as a labelled unordered tree. For any vertex v with outdegree d + T (v) we have a canonical bijection between the set of numbers [d + T (v)] = {1, 2, . . . , d + T (v)} and the offspring set v1, v2, . . . , vd + T (v). This allows us to interpret an enriched plane tree (T, β) as an enriched tree.
The following lemma provides a coupling that allows us to make use of the wealth of results for simply generated trees in order to study enriched trees.
Lemma 1.9. Let n with |A R [n]| ω > 0 be given. The outcome A R n = (A n , α n ) of the following procedure draws a random enriched tree from the set A R [n] with probability proportional to its ω-weight.
(1) Draw a simply generated plane tree T n of size n according to the weight sequence w.
uniformly at random and distribute labels by applying the transport function: In particular, by corresponding results on simply generated trees recalled in Section 2.1, we have that |A R [n]| > 0 implies that n | span(w) and conversely, if n | span(w) is large enough, then |A R [n]| > 0. The random enriched plane tree (T n , β n ) encodes all information about the enriched tree A n apart from the labeling. The vertices of enriched plane trees have unique coordinates which allow us to encode these objects as elements of a product space as follows.
Consider the countable metric space X := { * , ∞} n R[n] such that the set X \ {∞} is equipped with the discrete metric and the space X is the corresponding one-point compactification. Clearly X is a compact polish space and so is the countable product X V∞ .
An R-enriched plane tree (T, β) may be encoded as an element of X V∞ by setting β(v) := * for all vertices v ∈ V ∞ \ V (T ). By abuse of notation we will often just write (T, β) ∈ X V∞ . Let A ⊂ X V∞ denote the subset of all R-enriched plane trees that may have vertices with infinite degree. We do not require the offspring set of such a vertex v to be endowed with an additional structure and set β(v) := ∞. The subspace A is closed and hence also compact and a Polish space, see the proof of Theorem 1.10 in Section 3.2 for details.
We may now state our next main theorem which ensures the local convergence of our model of random enriched trees. Janson [Jan12,Thm. 7.1] showed the local convergence of simply generated trees (i.e. the case R = SEQ) and our proof builds on this result. Theorem 1.10. Let (T n , β n ) denote the random R-enriched plane tree from Lemma 1.9. We define the random modified R-enriched plane tree (T ,β) as follows.
(1) LetT ∈ T be the modified Galton-Watson tree defined in Theorem 2.1 that corresponds to the weight sequence (ω k ) k .
Then (T n , β n ) converges in distribution towards (T ,β) in the metric space A.
In particular, the R-structure of the root converges in distribution.
Corollary 1.11. Letting o denote the root of the simply generated tree T n it follows that the random R-structure β n (o) converges in distribution toβ(o).
Depending on the weight sequence we may show a stronger form of convergence.
Lemma 1.12. Suppose that the weight sequence w has type Iα. Then for any subset E ⊂ ∞ k=0 R[k] we have that P(β n (o) ∈ E) → P(β(o) ∈ E) as n ≡ 1 mod span(w) tends to infinity. We cannot expect Lemma 1.12 to hold for arbitrary weight sequences. For example, if it would also hold for weight-sequences of type II, then by the discussion below it would follow that the uniform rooted planar graph P n with n vertices has the property, that the limit probabilities d k = lim n→∞ P(root degree(P n ) = k) sum up to a value strictly smaller than 1. But they do sum up to 1, due to results by Drmota, Giménez and Noy [DGN11].
1.3.1. Applications to random graphs. As in Section 1.2.1, let C denote a block-class of connected graphs and B its subclass of graphs that are two-connected or are a single edge with its ends. Let γ be a weighting on the species B and let ω (resp. κ) denote the corresponding weightings on the species C • and C (resp. R = SET • B ). We consider the random connected graph C ω n drawn from the set C[n] with probability proportional to its weight, and the uniformly drawn graph C n .
Block sizes. The maximum block-size of the random graph C ω n is an important parameter which influences its geometric shape. We are going to show by a short argument that in many cases, including the case where C ω n is the uniform labelled planar graph, the maximum degree of the simply generated tree T n is a good approximation of this parameter. Hence known bounds and scaling limits for the maximum degree of simply generated trees Janson [Jan12, Ch. 9 and Thm. 19.34] and Jonsson and Stefánsson [JS11] also hold for the maximum block-size of C n . Recall the definition of the series φ and the numbers ν, τ and ρ φ from Section 2.1.
iii) Suppose w has type II and ω k ∼ ck −β for some constants c > 0 and β > 2. This includes the case that C ω n is the uniform labelled planar graph for which we have β = 5/2. Let α = β − 1 and c = c/φ(1). Then the following holds: We do not claim novelty for all results of Corollary 1.13, as this problem has already been studied by various authors. However, the elegant approach of building on results for simply generated trees instead of attacking the problem directly is new and, to the best knowledge of the author, the scaling limits of the block sizes in b) and c) are also new.
Panagiotou and Steger [PS10, Thm. 1.2] obtained by a detailed study of Boltzmann samplers that for any > 0 and sequence t n → ∞ the random labelled planar graph with n vertices has with high probability a unique giant block with (1 ± )cn vertices and the next largest block has size between n 2/3 / log(n) and n 2/3 t n .
Giménez, Noy and Rué [GNR13, Thm. 5.4] applied an elaborated analytical framework to obtain the strong result P(B (1) = k) ∼ n −2/3 f (k) uniformly for k = (1−ν)n+xn 2/3 an integer and x in a fixed compact interval. Here f denotes the density function of the distribution of the random variable X 3/2 from Corollary 1.13.
Drmota and Noy [DN13, Thm. 3.1] used analytic methods to show that if w has type Ia (and, for simplicity, span(w) = 1) then E[B (1) ] = O(log(n)) and if additionally the series B(z) satisfies the ratio test, then P(B (1) ≤ k) ∼ exp(− exp(log(n) − g(k))) uniformly for n, k → ∞ where g(k) is a function with g(k) ∼ Ck for some constant C > 0.
The block-tree. McDiarmid and Scott [MS14,p. 4] showed that with high probability any path in the graph C n passes through at most 5 n log(n) blocks and conjectured that the extra factor log(n) could be replaced by any sequence tending to infinity. By Lemma 3.4 there is a coupling with a simply generated tree T n such that the diameter D(T n ) and the maximum number D n of blocks along a path in C ω n differ by at most 1. In particular, if the weight sequence w has type Iα, then it follows that D n / √ n converges in distribution and the conjecture holds in this case. If w has type Iβ, it is reasonable to expect that D(T n )/ √ n converges in probability to zero, but this is still an open conjecture [Jan12, Conj. 21.5]. By results of Addario-Berry, Devroye and Janson [ABDJ13] we know that if w has type Iα, then there are constants C, c > 0 depending on w such that for all h and n it holds that P(H(T n ) ≥ h) ≤ C exp(−ch 2 /n). Janson [Jan12, Problem 21.8] posed the question, whether this holds for the remaining types of weight-sequences as well. If this question can be answered in the affirmative, then the conjecture for the diameter of the block-tree follows immediately.
In the following nongeneric case we may show a stronger bound. Recall the definition of the constant ν given in Section 2.1.
Corollary 1.14. Suppose that the weight-sequence w has type II and ω k ∼ ck −β for some constants β > 2 and c > 0. Choose a vertex of C ω n uniformly at random and let h n denote the maximum number of blocks along a path starting in that vertex. Then h n / log(n) converges to the constant log(1/ν) in the space L p for any p ≥ 1. In particular, this applies to the uniform random planar graph for which we have t = 5/2. This follows directly by combining Lemma 1.9 with a result due to Kortchemski [Kor, Thm. 4 and Prop 2.11] who showed in a more general setting that H(T n )/ log(n) converges towards the constant log(1/ν) in the space L p for any p ≥ 1. Local properties. The question what happens locally around a uniformly at random chosen root of the graph C n for large n has received some attention in recent literature. In particular, the limit distribution of the root-degree was studied in [BPS09, DGN11, PS11, BPS08] with a focus on planar graphs, outerplanar graphs, series-parallel graphs and arbitrary subcritical classes (i.e. the case that the weight-sequence w has type Ia). The results from this section provide new proofs for results in the subcritical case and a new probabilistic description of the limit distribution.
Let r denote a uniformly at random chosen root vertex of the graph C ω n . By Theorem 1.10 the random rooted graph (C ω n , r), encoded up to relabeling by an enriched plane tree (T n , β n ), converges weakly towards an infinite rooted graphĈ • , encoded by (T ,β). In particular, if B 1 , . . . , B s is a given list of blocks from the class B, then the probability p n , that the root vertex r lies in precisely s blocks B 1 , . . . , B s , with B i B i for all i, converges towards the probability p that the root ofĈ • has this property. Consequently, the number N (n) of vertices of C n having this property satisfies E[N (n)] ∼ np as n ≡ 1 mod span(w) becomes large. If the weight-sequence w has type Iα, then we obtain a much stronger result. There is a unique label t of the SEQ • B -objectβ(o) of the root o, that corresponds to the second vertex of the spine of the treeT . Thusβ(o) together with the distinguished label t form an R • -object. This pointed object satisfies a weighted Boltzmann-distribution ΓR • (y) i.e. any object m from the set k∈N 0 R • [k] gets drawn with probability proportional to y |m| |m|! B γ B with the index B ranging over the blocks of m. The parameter y is given by y = C • (ρ C ) with ρ C the radius of convergence of the exponential generating series C • (z). The distribution of β(o) is given by the distribution of the R-object corresponding to the pointed object ΓR • (y).
The rules for operations on species imply that R • (SET • B ) · B • . Lemma 1.12 now reads as follows.
Corollary 1.15. Suppose that the weight sequence w has type Iα. Let E ⊂ ∞ k=0 R[k] be a subset that is closed under relabelling. Then the probability, that the SET • B -object corresponding to a uniformly at random chosen root of the graph C ω n belongs to E, converges as n becomes large to the probability that the weighted Boltzmann-distributed object Γ(SET•B )B • (y) belongs to E.
For example, let H be set of two-connected graphs and E be any of the following properties: "r has degree k", "there are precisely k circles in the graph C n containing the vertex r" or, more generally, "C n contains precisely k isomorphic copies of graphs from H as subgraphs that contain the vertex r". Then the probability, that the root of the graph C • n has property E, converges towards the probability, that the inner root of Γ(SET • B )B • (y) has this property. In particular, the limit d k = lim n→∞ P(the root of C • n has degree k) exists and (d k ) k is a probability distribution given by the root-degree of Γ(SET • B )B • (y) interpreted as a graph. The existence of a limit distribution for the root degrees in the case that C is subcritical and aperiodic (i.e. w has type Ia and span(w) = 1) was previously shown by Bernasconi, Panagiotou and Steger [BPS09] using different methods. The probabilistic interpretation of the limit distribution as a parameter of a Boltzmann distributed graph is new, however.

Preliminaries
2.1. Simply generated trees. We recall required notions and results regarding the convergence of simply generated trees following closely Janson [Jan12].
Any plane tree T can be viewed as subtree of the Ulam-Harris tree U ∞ and is uniquely determined by its sequence of outdegrees (d + We endow the set N 0 with a compact topology as the one-point compactification of the discrete space is a compact Polish-space since it is the product of countably many such spaces. We let T ⊂ N V∞ 0 denote the subspace of trees which may have nodes with infinite outdegree. The subset T is closed and hence also compact. Let ξ be a random variable on N 0 with average value µ := E[ξ] ≤ 1 and let (π k ) k≥0 denote its distribution. The modified Galton-Watson treeT ∈ T is defined in [Jan12, Ch. 5] as follows. Any vertex is either normal or special and we start with a root vertex that is declared special. Normal vertices have offspring according to an independent copy of ξ and special vertices have offspring (outdegree) according to an independent copy of the random variableξ with distribution given by All children of a normal vertex are declared normal and if a special node gets an infinite number of children all are declared normal as well. When a special vertex gets finitely many children all are declared normal with one uniformly at random chosen exception which is declared special. The special vertices form a path which is called the spine of the treeT . Note that if µ < 1 (the subcritical case) thenT has almost surely a finite spine ending with an explosion. If µ = 1 thenT is almost surely locally finite and has an infinite spine.
Let w = (ω k ) k∈N 0 with ω k ∈ R ≥0 for all k denote a weight sequence satisfying ω 0 > 0 and ω k > 0 for some k ≥ 2. Recall that the support of w is defined by supp(w) = {k | ω k > 0} and the span by span(ω) is the greatest common divisor of the support.
By [Jan12,Cor. 15.6], we know that if there is a plane tree with n vertices having a positive ω-weight then span(ω) | n and conversely, if n is large enough, then span(ω) | n implies the existence of such a tree.
Consider the power series defined on [0, ρ φ [ is finite, continuous and strictly increasing. We define If ρ φ = 0, set ν := ψ(0) := 0. The result for the local convergence of simply generated trees may be stated as follows.
Define the probability distribution (π k ) k on N 0 by and letT denote the corresponding modified Galton-Watson tree. Then, as n ≡ 1 mod span(w) tends to infinity, we have that in the metric space T. The expected value and variance of the distribution (π k ) k are given by µ = ψ(τ ) = min(ν, 1) and σ 2 = τ ψ (τ ) ≤ ∞.
Equivalently, we may define τ as the unique number in the interval [0, ρ φ ] satisfying the equation τ φ (τ ) = φ(τ ). The generating function Z(z) = ∞ n=0 Z n z n with Z n = T ∈Tn ω(T ) is the unique power series satisfying This follows from [Jan12, Rem. 3.2] and the Lagrange inversion formula. Let ρ Z denote its radius of convergence. By [Jan12, Ch. 7] we have that 0 ≤ ρ Z < ∞ and Moreover, ρ Z = 0 ⇔ ρ φ = 0 ⇔ τ = 0 and it holds that It is convenient to partition the set of weight sequences into the three cases I) ν ≥ 1, II) 0 < ν < 1 and III) ν = 0. The case I) may be subdivided in mutually exclusive cases by either Ia) ν > 1 and Ib) ν = 1, or Iα) ν ≥ 1 and σ < ∞ and Iβ) ν = 1 and σ = ∞. In the cases I) and II) the simply generated tree with n vertices is distributed like a Galton-Watson tree conditioned on having size n with offspring distribution (π k ) k .
2.2. Gromov-Hausdorff convergence. We briefly introduce the Gromov-Hausdorff metric required for the scaling limits in Theorems 1.1 and 1.3. Given two compact metric spaces (X, d) and (Y, d) a correspondence between X and Y is a subset R ⊂ X × Y having the property that for any x ∈ X there is a y ∈ Y with (x, y) ∈ R, and conversely for any y ∈ Y there is a x ∈ X with (x, y) ∈ R. The distortion of the correspondence is defined as the supremum dis(R) = sup{|d X (x 1 , x 2 ) − d Y (y 1 , y 2 ) | (x 1 , y 1 ), (x 2 , y 2 ) ∈ R}.
We define the Gromov-Hausdorff distance between the metric spaces X and Y by with the index R ranging over all correspondences between X and Y . We require the factor 1 2 in order to stay consistent with the more common definition of the Gromov-Hausdorff distance via the Hausdorff distance of embeddings of X and Y in a common metric space. See for example [BBI01, Thm. 7.3.25]. The set K of isometry classes of compact metric spaces equipped with the Gromov-Hausdorff distance is known to be a Polish space [BBI01]. showed that there exist constants C, c > 0 such that the following tail-bound for the height H(T n ) holds: for all n and h ≥ 0. Recall that the lexicographic depth-first-search (DFS) of the plane tree T n is defined by listing the vertices in lexicographic order v 0 , v 1 , . . . , v n−1 and defining the queue (Q i ) 0≤i≤n by Q 0 = 1 and the recursion Compare with Figure 5, in which the numbers Q i are adjacent to the vertices v i . We may also consider the reverse DFS (Q i ) 0≤i≤n as the DFS of the tree obtained from T n by reversing the ordering on each offspring set. Then (Q i ) i and (Q i ) i agree in distribution and by [ABDJ13, Ineq. (4.4)] there are constants C, c > 0 such that for all n and x ≥ 0. Given a vertex v of T n let j and k denote the corresponding indices in the DFS and reverse DFS. In particular, v = v j in the lexicographic ordering. Then with the sum index u ranging over all ancestors of v and with h Tn (v) denoting the height of the vertex v.

2.4.
Operations on weighted species. Following Joyal [Joy81] and Bergeron, Labelle and Leroux [BLL98], we briefly recall several notions and facts about weighted species to the extend required in the present paper. To any weighted species F ω we associate its exponential generating series and ordinary generating series defined by We which, by abuse of notation, we also denote by ω. It is given by ω(m, σ) = ω(m)s σ 1 1 s σ 2 2 · · · with σ i denoting the number of cycles of length i of the permutation σ. The cycle index sum Z F ω of F ω is defined to be the exponential generating series of the weighted species Sym(F) ω . The generating series are related by andF ω (z) = Z F ω (z, z 2 , z 3 , . . .).
Let G ν be another weighted species. We say F and G are isomorphic, denoted by F G, if there exists a natural isomorphism between these functors. If the natural isomorphism preserves weights we write F ω G ν . In this case the cycle index sums and hence also the other two generating series of F ω and G ν coincide.
The main point of considering symmetries is that if we sample a symmetry (m, σ) from the set Sym(F)[n] with probability proportional to its weight, then the isomorphism type t(m) of the F-object m satisfies P(t(m) = t) = ω(t)/ s∈F [n] ω(s) for any unlabelled F-object t of size n. This follows from basic properties of group operations.
Given weighted species F ω and G ν we may combine these species to form a new species in several ways. Their product is given by with the index ranging over all ordered 2-partitions of U , i.e ordered pairs of (possibly empty) disjoint sets whose union equals U . The transport of the product along a bijection is defined componentwise. We have a canonical weighting on the product given by µ(F, G) = ω(F )ν(G). The corresponding cycle index sum is given by Z (F ·G) µ = Z F ω Z G ν . Let (F ω i i ) i∈I be a family of weighted species such that for any finite set U only finitely many indices i with F i [U ] = ∅ exist. Then the sum i∈I F ω i i is a species defined by with a canonical weighting µ given by µ(m) = ω i (m) for any i and m ∈ F i [U ]. The corresponding cycle index sum is given by Z with * U referring to an arbitrary fixed element not contained in the set U . The weight of a derived object m ∈ F [U ] is given by ω U ∪{ * U } (m). The transport along a bijection σ : U → V is done by applying the transport F[σ ] of the bijection σ : The corresponding cycle-index sum is given by Z F = ∂ ∂s 1 Z F ω . The pointed species F • is given by the product of weighted species with X denoting the species given by single object of size 1 which has weight 1. In other words, a F • -object is pair of a F-object and a distinguished label which we call the root of the object. If G[∅] = ∅, then we may form the composition defined by with the index π ranging over all unordered partitions of the set U . Here the transport along a bijection σ is done by applying the induced map of partitionsσ : π →π to the F-object and the restrictions σ| Q , Q ∈ π to the G-objects. For example, the species A of rooted trees satisfies an isomorphism A X · (SET • A) with X denoting the species given by a single object with size 1. There is a canonical weighting µ on the composition given by µ(π, m, (G Q ) Q∈π ) = ω(m) Q∈π ν(Q). The cycle index sum of the composition is given by 2.5. Decomposition of symmetries. We are going to need detailed information on the structure of the symmetries of the composition F • G. The following is a standard decomposition given in [Joy81,BLL98,BFKV11]. Let U be a finite set. Any element of Sym(F • G) [U ] consists of the following objects: a partition π of the set U , a F-structure F ∈ F[π], a family of G-structures (G Q ) Q∈π with G Q ∈ G[Q] and a permutation σ : U → U . We require the permutation σ to permute the partition classes and induce an automorphismσ : π → π of the F-object F . Moreover, for any partition class Q ∈ π we require that the restriction σ| Q : Q → σ(Q) is an isomorphism from G Q to G σ(Q) . For any cycleτ = (Q 1 , . . . , Q ) of σ it follows that for all i we have σ (Q i ) = Q i and the restriction σ | Q i : Q i → Q i is an automorphism of G Q i . Conversely, if we know (G Q 1 , σ | Q 1 ) and the maps σ| Q i = (σ| Q 1 ) i for 1 ≤ i ≤ − 1, we can reconstruct the G-objects G Q 2 , . . . , G Q and the restriction σ| Q 1 ∪...∪Q . Here any k-cycle (a 1 , . . . , a k ) of the permutation σ | Q 1 corresponds to the k -cycle (a 1 , σ(a 1 ), . . . , σ −1 (a 1 ), a 2 , σ(a 2 ), . . . , σ −1 (a 2 ), . . . , a k , σ(a k ), . . . , σ −1 (a k )) of σ| Q 1 ∪...∪Q . Thus any cycle ν of σ corresponds to a cycle of the induced permutationσ whose length is a divisor of the length of ν. This implies that any F • G-symmetry is isomorphic to a symmetry ((π, F, (G Q ) Q∈π ), σ) constructed in the following way. Choose a F-symmetry (m, σ m ). For any cycle τ of the permutation σ m choose a G-symmetry (G τ , σ τ ) and let Q τ denote its set of labels.  ((a, b), . . . , (a, τ −1 (b)), (ν(a), b), . . . , (ν(a), τ −1 (b)), . . . , (ν k−1 (a), b), . . . , (ν k−1 (a), τ −1 (b))).
Then the product σ of all composed cycles is an automorphism of the F • G-structure C. The composed cycles are pairwise disjoint, hence it does not matter in which order we take the product. Note that σ does not depend on the choice of the a's but different choices of the b's result in a different automorphism σ. More precisely, if for a given cycle τ we choose τ (b) instead of b, then the resulting automorphism is given by the conjugation (id, τ )σ(id, τ ) −1 instead of σ. But (id, τ ) is an automorphism of the F • G-structure C, hence the resulting symmetry (C, (id, τ )σ(id, τ ) −1 ) is isomorphic to (C, σ). This implies that the isomorphism type of (C, σ) does not depend on the choices of the a's and b's. In particular, the isomorphism A R X · R(A R ) allows us to recursively decompose a symmetry (A, σ) of an A R -enriched tree into a Sym(R)-enriched tree, compare with Figure 7. By doing so we lose some information about σ, but not about the isomorphism type of (A, σ).
2.6. Deviation inequalities. We will make frequent use of the following inequality.

Proofs of the main results
3.1. Proofs of Section 1.3. We start with a proof for the coupling of random enriched trees with simply generated trees. This Lemma lies at the core of all further results.
Proof of Lemma 1.9. Let T n denote the set of plane trees with n vertices and Z n = T ∈Tn ω(T ) the partition function of the weight sequence (ω k ) k . Let A denote the species of rooted unordered trees. Every unordered rooted tree A ∈ A[n] corresponds to v∈V (A) d + A (v)! ordered trees (with labels in the set [n]) and every plane tree corresponds to n! ordered (labelled) trees. Hence This concludes the proof.
Next, we are going to proof the local convergence of our model of random enriched trees.
Proof of Theorem 1.10. The proof is inspired by the proof of [Jan12, Thm. 7.1]. However, we may build on the results for simply generated trees instead of repeating all arguments. By construction, the set of enriched trees A is a subset of the compact product space X V∞ with X = { * , ∞} n R[n]. We are going to argue that A is also compact. For any vertex v ∈ V ∞ and integers i > k ≥ 0 set

Then each subset U v,k,i is open. Thus the subspace
is closed and hence compact.
Since A is compact, any sequence of random enriched plane trees has a convergent subsequence. In particular, the sequence (T n , β n ) of random enriched plane trees converges towards a limit object (T ,β) along a subsequence (n k ) k . We are going to show that (T ,β) =T .
But T n =T .
Moreover, in order to show (T ,β) The set X V is countable, hence this is equivalent to =T it suffices to consider the case because otherwise both sides of Equation ( * ) equal zero. In particular, since the treeT has almost surely at most one vertex with infinite degree, we may assume that at the number of vertices v ∈ V , with R v = ∞, equals one or zero. Case a): Moreover, by Lemma 1.9 and the definition of (T ,β) it holds that Hence Equation ( * ) holds in this case. Case b): R u = ∞ for precisely one u ∈ V . By a similar argument as in case a) it follows that for all K ≥ 1 Letting K tend to infinity yields Equation ( * ).
Next, we are going to prove the strong type of convergence around the root, if the weight sequence has type Iα.
Proof of Lemma 1.12. For any k ≥ 1 set p k = |E ∩ R[k]| κ /|R[k]| κ . Then it holds that Hence we have to show that P(β n (o) ∈ E) tends to ∞ k=1 p k kπ k as n becomes large. The weight sequence w has type Iα by assumption, hence by construction of the limit treeT we have that dT (o) < ∞ almost surely. Since d + Tn (o) converges weakly to dT (o) it follows that Let ξ denote a random nonnegative integer with distribution (π k ) k and let (ξ i ) i∈N a family of independent copies of ξ. Moreover, set S = i=1 (ξ i − 1). Then , see for example Janson [Jan12, Rem. 15.8]. Since E[ξ] = 1 and ξ has finite nonzero variance σ 2 , the local limit theorem for lattice distributed random variables ensures that )| tends to zero as n becomes large. Hence the ratio P(S n−1 = −k)/P(S n = −1) converges to 1 uniformly for all integers 1 ≤ k ≤ log(n). Hence sup 1≤k≤log(n) |P(d Tn (o) = k) − kπ k | → 0 and thus P(β n (o) ∈ E) tends to ∞ k=1 p k kπ k as n becomes large. For our applications to random planar graphs, we need to check that they fit into our setting.
Lemma 3.1. Suppose that C is the class of connected planar graphs and B the subclass of all two-connected planar graphs and graphs that consists of a single edge with its ends. Let w = (ω k ) k denote the corresponding weight sequence given by ω k = |(SET • B )[n]|/n!. Then w has type II and there is a constant c with ω k ∼ ck −5/2 as k becomes large.
Proof. By enumeration results given in [GN09], there are constants ρ C , ρ B , C, B > 0 such that the coefficients of the generating series C • (z) and B (z) are asymptotically given by [GN09,Lem. 6] the function B (z) has a unique singularity on the circle |z| = ρ B , which has type 3/2. Hence the same holds for exp(B (z)). By standard singularity analysis [FS09,Ch. 6], it follows that ω k ∼ ck −5/2 for some constant c > 0 as k becomes large. Moreover, by [GN09, Claim 1] we know that (See Section 2.1 for the definition of the parameter ν.) I.e. the weight sequence w has type II. This concludes the proof.
It remains to provide a proof of Corollary 1.13. To this end we need a small lemma regarding random set partitions.
Lemma 3.2. Let F γ be a weighted species satisfying f n /n! := [z n ]F γ (z) ∼ an −β r −n and s n /n! := [z n ](SET • F γ )(z) ∼ bn −β r −n for some constants β > 1 and a, b, r > 0. Let X n denote the size of the largest component of a random object from the set (SET • F)[n] drawn with probability proportional to its weight. Then for any > 0 there exist constants C, δ > 0 such that for all n Proof. If X n ≤ n/4, then we may divide the set of components into two parts with n/2 − x and n/2 + x vertices for some 0 ≤ x ≤ n/4. The probability for this event is bounded by x n n/2 − x s n/2−x s n/2+x /s n with the sum index x ranging over all multiples of 1/2 in the interval [0, n/4] such that n/2+x is an integer. Since holds uniformly for all x, it follows that there is a constant C 0 > 0 such that P(X n ≤ n/4) ≤ C 0 n 1−β for all n. For any integer k we have that P(X n = k) is bounded by n k f k s n−k /f n . Hence there is a constant C such that P(X n = k) ≤ C (k(n − k)/n) −β for all k. It follows that for any 1 > 1/β there is a constant C 1 > 0 such that P(n/4 ≤ X n ≤ n − n 1 ) ≤ C 1 n 1− 1 β for all n. Given 2 > 1 /β we may repeat the same arguments to obtain the bound P(n − n 1 ≤ X n ≤ n − n 2 ) ≤ C 2 n 1 − 2 β for some constant C 2 > 0. We may repeat this argument arbitrarily many times and choose the i arbitrarily close to 1/β i . Hence for any > 0 there are constants C, δ > 0 with P(X n ≤ n − n ) ≤ Cn −δ for all n.
Proof of Corollary 1.13. Recall that by Lemma 1.9 the connected random graph is generated by drawing a simply generated tree T n and then partitioning the offspring sets into sets of derived blocks. Let Y (i) denote the ith largest outdegree of the simply generated tree T n . If we replace the maximum block-sizes B (i) with the maximum outdegrees Y (i) in Corollary 1.13, then all claims hold by [Jan12,Thm. 19.34 and Ch. 9] and [JS11]. Hence all we have to show is that these extremal parameters are sufficiently close. To this end, we are going to argue, that the partitions of large offspring sets have giant components that correspond to large blocks. By the construction of the coupling in Lemma 1.9 we have that B (1) ≤ Y (1) + 1, hence claim ii) and the upper bound in claim i) follow immediately. By Lemma 3.2 it follows that for any > 0 there are positive constants δ, C such that . This suffices to prove the remaining claims: Suppose that w has type Ia and ω 1/k k converges. If 1/ log(ρ φ /τ ) = 0, then Y (1) / log(n) p −→ 0 and since B (1) ≤ Y (1) + 1 it follows that If 1/ log(ρ φ /τ ) > 0, then w.h.p. Y (1) ≥ log(n)/(2 log(ρ φ /τ )) and it follows by Inequality ( * ) that Under the assumptions of claim iii) we have Y (1) ≥ 1−ν 2 n with high probability and hence it follows by Inequality ( * ) that for any > 0 we have that This proves part a).
In order to prove parts b) and c) let j ≥ 2 be given and set = 1/(2α). We are going to argue that with high probability it holds for all 1 ≤ i ≤ j that This implies that |Y (i) − B (i) |/n 1/α converges in probability to zero and we are done. In the following we say that a block B belongs to a vertex v, if v ∈ B and B := B −v consists only of offspring vertices of v. Using a similar argument as for Inequality ( * ), we may apply Lemmas 1.9 and 3.2, and the scaling limits for the Y (i) in order to show that with high probability there are distinct vertices v 1 , . . . , v j and distinct blocks B 1 , . . . , B j such that for all 1 ≤ i ≤ j the following holds.
(1 − ν)n ± n 1/α log n and n 1/α / log n ≤ Y (k) ≤ n 1/α log n for 2 ≤ k ≤ j. In order to show the lower bound in Inequality ( * * ), observe that for any integer 1 ≤ i ≤ j the vector (B (1) , . . . , B (i) ) dominates coefficient-wise the sorted list sort(|B 1 |, . . . , |B i |) (in descending order). This is due to the fact that the blocks B i are distinct. Hence it follows by 2) that For the upper bound, suppose that there is an index 1 ≤ ≤ j with B ( ) > Y ( ) +1. Then ≥ 2 and each block of the graph with size at least B ( ) must belong to a vertex with outdegree strictly larger than Y ( ) . All vertices with outdegree strictly larger than Y ( ) are contained in the set {v 1 , . . . , v −1 }. Hence there are only − 1 possible candidates of vertices, but at least blocks which belong to one of these vertices. Hence there must be an index 1 ≤ i ≤ − 1 and two blocks with size at least B ( ) which belong to the vertex v i . By 2), 3) and the choice of it follows that Hence the block B i is the only block that belongs to v i and may have size greater than or equal to B , a contradiction. Hence it must hold that B ( ) ≤ Y ( ) for all 1 ≤ ≤ j. This completes the proof of Inequality ( * * ) and hence the proof of b) and c). This concludes the proof.

The labelled setting.
Proof of Theorem 1.1. Consider the coupling of the random enriched tree A R n with the enriched plane tree (T n , β n ) given in Lemma 1.9. By assumption, the weight sequence w has type Ia. Hence by the discussion in Section 2.1 the simply generated tree T n is distributed like a critical Galton-Watson tree T conditioned on having size n, such that the offspring distribution (π k ) k has finite exponential moments. In particular, as n ≡ 1 mod span(w) tends to infinity, we have that σT n /(2 √ n) converges towards the CRT T e with σ 2 denoting the variance of the offspring distribution.
For any finite set U and R-structure R ∈ R[U ] let η R denote the δ R -distance from the * U point to a uniformly at random chosen label from U . Moreover, let η denote the random number given by choosing an integer k according to the distribution (kπ k ) k , choosing a random R-structure R from R[k] with probability proportional to its κ-weight and setting η = η R . By assumption, for any R-structure R the random variable η R is bounded by the sum of |R| copies of a random variable χ ≥ 0 having finite exponential moments. Since the distribution (kπ k ) k also has finite exponential moments, it follows that η has finite exponential moments.
We are going to show that the Gromov-Hausdorff distance between E[η]T n / √ n and X n / √ n converges in probability to zero. This immediately implies that −→ T e and we are done. Let s > 1 and t > 0 be arbitrary constants and set s n = log(n) s and t n = n t . Let > 0 be given and let E 1 denote the event that there exists a vertex v ∈ T n and an ancestor u of v with the property that We are going to show that with high probability none of the events E 1 and E 2 takes place. This suffices to show the claim: Take s = 2 and t = 1/4 and suppose that the complentary events E c 1 and E c 2 hold. Given vertices a = b let x denote ther lowest common ancestor in the tree T n . If x ∈ {a, b} then we have If x = a, b, then let a denote the offspring of x that lies on the T n -path joining a and x and likewise b the offspring of x lying on the path joining x and b. Hence we have that By property E c 2 and the triangle inequality it follows that |R| = −R ≤ 2n 1/4 . Thus, regardless whether x ∈ {a, b}, it holds that Tn (a, x).
Otherwise, if d Tn (a, x) < log(n) 2 then it follows by property E c 2 that d Xn (a, x) ≤ n 1/4 and thus |d Xn (a, x) − E[η]d Tn (a, x)| ≤ Cn 1/4 for a fixed constant C that does not depend on n or the points a and x. It follows that with D(T n ) denoting the diameter. Thus holds with high probability. Since we may choose arbitrarily small, and D(T n )/ √ n converges in distribution (to a multiple of the diameter of the CRT), it follows that d GH (X n , E[η]T n ) → 0 in probability and we are done.
It remains to show that the events E c i hold with high probability. To this end, recall the construction of the modified Galton-Watson treeT from Section 2.1 that corresponds to the distribution (π k ) k . Given ≥ 0 we may consider the truncated versionT ( ) which has a finite spine of length . At the top of the spine the special node becomes normal and reproduces normally. We call this vertex the outer root, i.e.T ( ) is a random pointed plane tree. This construction was introduced in [ABDJ13, Ch. 3] and has the property, that for any plane tree T with a distinguished vertex r that has height = h T (r) we have that P(T ( ) = (T, r)) = P(T = T ). This is due to the fact that the probability, that a special node has offspring of size k and precisely the ith child is declared special, is given by kπ k /k = π k . For any vertices v of T and , each with probability proportional to its κ-weight. In particular, (T n , β n ) is distributed like (T , β) conditioned on having n vertices. For any pointed enriched plane tree ((T, r), γ) we have that P((T ( ) ,β ( ) ) = ((T, r), γ)) = P((T , β) = (T, γ)).
For each R ∈ n≥0 R[n] let (δ i R ) i∈N 0 be a family of independent copies of δ R . Given an Renriched plane S = (T, γ) we may form the family (δ S (v)) v∈T of random metrics by traversing bijectively the vertices of T in a fixed order, let's say in depth-first-search order, and assigning to each vertex v the "leftmost" unused copy from the list (δ 1 γ(v) , δ 2 γ(v) , . . .). The metrics can be patched together to a metric d S on the vertex set of the plane tree T just as described in Section 1.2.
We may assume that all random variables considered so far are defined on the same probability space and that the metric d Xn of X n coincides with the metric d (Tn,βn) . Given (δ i R ) R,i let H denote the finite set of R-enriched plane trees of size n such that the event E 1 takes place if and only if (T n , β n ) ∈ H. By the definition of the event E 1 for any H = (T, γ) ∈ H we may fix a vertex v H of H = (T, γ) having the property that there exists an ancestor u with Let H denote the height h T (v H ). The probability for the critical Galton-Watson tree T to have size n is Θ(n −3/2 ) and hence the conditional distribution of the event E 1 given (δ i R ) R,i equals Let v 0 , . . . , v denote the spine ofT ( ) , i.e. v is the outer root, v 0 is the inner root, and (v 0 , . . . , v ) is the directed path connecting the roots. It follows that the probability for the event E 1 is bounded by But the d (T ( ) ,β) -distance between two spine vertices v i and v j is distributed like the sum η 1 + . . . + η |i−j| of independent copies (η i ) i of η. We know that η has finite exponential moments and hence, by the deviation inequality in Lemma 2.2, the bound ( * ) converges to zero as n ≡ 1 mod span(w) tends to infinity. Thus the event E 1 holds with high probability.
By the same arguments we may bound the probability for the event E 2 by which also converges to zero. This concludes the proof.
Proof of Lemma 1.2. It suffices to show that there are constants C, c, N > 0 such that for all n ≥ N and h ≥ √ n we have that P(H(X n ) ≥ h) ≤ C(exp(−ch 2 /n) + exp(−ch)). Recall the coupling in Lemma 1.9 of the random graph A R n with a simply generated tree T n sharing the same vertex set. The weight sequence w has type Iα by assumption, hence T n is distributed like a critical Galton-Watson tree conditioned on having n vertices with the offspring distribution having finite nonzero variance.
For any vertex v set (v) = u d + Tn (u) with the sum index u ranging over all ancestors (not equal to v) of the vertex v in the plane tree T n . Let s > r > 0 be constants. Given h ≥ √ n let E h denote the event that H(X n ) ≥ h. Clearly with E h 1 the event that H(T n ) ≥ rh, E h 2 the event that H(T n ) ≤ rh, and there exists a vertex v with (v) ≥ sh and E h 3 the event that H(T n ) ≤ rh, (v) ≤ sh for all vertices v and H(X n ) ≥ h. We are going to show that if we choose r and s sufficiently small then each of these events is sufficiently unlikely. By the tail bound (2.1) for Galton-Watson trees it follows that there are constants C 1 , c 1 > 0 such that P(E h 1 ) ≤ C 1 exp(−c 1 h 2 /(r 2 n)). In order to bound the probability for the event E h 2 suppose that we are given a vertex v with h Tn (v) ≤ rh, and (v) ≥ sh. Let (Q i ) i and (Q i ) i denote the DFS and reverse DFS queues as defined in Section 2.3 and let j and k denote the indices corresponding to the vertex v in the lexicographic ordering of T n and its mirror image, respectively. By Equality (2.3) it follows that Q j + Q k = 2 + (v) − h Tn (v) ≥ (s − r)h and hence Q j ≥ (s − r)/2 or Q k ≥ (s − r)/2. It follows by Inequality 2.2 that P(E h 2 ) ≤ C 2 exp(−c 2 h 2 /n) for some constants C 2 , c 2 > 0 that do not depend on n or h.
We assumed that there is a real-valued random variable χ ≥ 0 such that for any R-structure R the diameter of the metric δ R is bounded by the sum of |R| independent copies χ R 1 , . . . , χ R |R| of χ. In particular, for any vertex v of X n the height h Xn (v) is bounded by the sum of (v) independent copies of χ. It follows that P(E h 3 ) ≤ nP(χ 1 + . . . + χ sh ≥ h) with (χ i ) i∈N a family of independent copies of χ. By the inequality in Lemma 2.2 it follows that there are constants λ, c > 0 such that . We assumed that h ≥ √ n and n ≥ N , hence we may take s sufficiently small and N sufficiently large (depending only on λ and c and thus not depending on h or n) such that there are constants C 3 , c 3 with for some constants C 4 , c 4 > 0 and we are done.
3.2.2. The unlabelled setting. Again we are going to study a coupling of the random metric space Y n with a conditioned Galton-Waltson tree and show that the Gromov-Hausdorff between Y n and a constant multiple of the Galton-Watson tree converges in probability to zero as n tends to infinity. The Galton-Watson tree will be given by the fixpoints of an A Rsymmetry drawn randomly according to a Boltzmann-distribution. Our main tools will be the framework of Pólya-Boltzmann samplers introduced by Bodirsky, Fusy, Kang and Vigerske [BFKV11] and a decomposition of pointed enriched trees due to Labelle [Lab81]. We start with a basic observation on enumerative properties. Lemma 3.3. Suppose that the ordinary generating seriesÃ ω R (z) has radius of convergence ρ > 0 and that the weighted species R κ has structures with positive weight of size 0 and k for some k ≥ 2. Then ρ and the sumÃ ω R (ρ) are both finite. If additionally the function E(z, u) = zZ R κ (u,Ã ω R (z 2 ),Ã ω R (z 3 ), . . .) satisfies E(ρ + ,Ã ω R (ρ) + ) < ∞ for some > 0, then the nth coefficient ofÃ ω R (z) is asymptotically given by ρ −n n −3/2 as n ≡ 1 mod span(w) tends to infinity. Moreover, we have that E u (ρ,Ã ω R (ρ)) = 1.
. By the assumptions on R κ it follows that there are constants a, b > 0 and k ≥ 2 with the property that for all 0 ≤ x < ρ it holds that . This implies that lim x↑ρÃ ω R (x) < ∞ and hence, by nonnegativity of coefficients,Ã ω R (ρ) < ∞. In particular we may apply a general enumeration theorem by Bell, Burris and Yeats [BBY06,Thm. 28] which implies the asymptotic behaviour of the coefficients of the power series A ω R (z). By Pringsheim's theorem the functionÃ ω R (z) cannot be analytically continued in a neighbourhood of ρ, hence by the implicit function theorem it must hold that the function Hence E u (ρ,Ã ω R (ρ)) = 1. (Compare with the proof of [BBY06,Cor. 12].) We are going to use the following recursive procedure illustrated in Figure 8 in order to sample random A R -symmetries according to a Boltzmann-distribution.
Here σ i (o) denotes the number of i-cycles of the permutation σ(o). 2. Recall that we count fixpoints of permutations as 1-cycles. For each cycle τ of the permutation σ(o) draw an independent copy (T τ , β τ ) of the recursively called sampler ΓS(x |τ | ) with |τ | ≥ 1 denoting the length of the cycle. For each atom a of the cycle τ make an identical copy (T a , β a ) of (T τ , β τ ). This procedure terminates almost surely. As described in Section 2.5 the resulting enriched plane tree (T , β) corresponds to a symmetry on the vertex set of the tree T . Let ΓZ A ω R (x) denote the result of relabelling this symmetry uniformly at random with labels from the set [n], with n denoting the number of vertices of the tree T . Then for any symmetry (A, σ) from the set k≥0 Sym(A R )[k] it holds that Proof. The rules for the construction of (recursive) Pólya-Boltzmann samplers [BFKV11, Ch. 5] can easily be extended to the weighted setting. This ensures, without the need of any further calculation, that the procedure ΓS(x) terminates almost surely and that ΓZ A ω R (x) satisfies the above Boltzmann-distribution. Note that if we condition the sampler ΓZ A ω R (ρ) on having size n, then any symmetry from Sym(A R )[n] gets drawn with probability proportional to the ω-weight of its A R -object. By the discussion in Section 2.5 it follows that the isomorphism class of this A R -object is distributed like the random enriched treeÃ R n . By Lemma 3.3 we know that under additional assumptions on the function E(z, u) the probability for the event |ΓZ A ω R (ρ)| = n is asymptotically Θ(n −3/2 ), i.e. it is only polynomially small. This will make it easy to study the conditioned sampler. Suppose that ρ > 0 and let (T , β) be drawn according to the sampler ΓS(ρ). The vertices of T that correspond to fixpoints of the symmetry ΓZ A ω R (ρ) form a subtree T f ⊂ T containing the root. Note that by the discussion in Section 2.5 the fixpoints correspond precisely to the vertices in which the sampler ΓS calls itself with parameter ρ (as opposed to parameter ρ i for some i ≥ 2). The plane tree T f is distributed like a Galton-Watson tree with offspring distribution ξ having probability generating function For any vertex v of T f let f (v) denote the offspring of v that correspond to fixpoints and let F (v) denote the forest of the maximal subtrees of T whose roots are offspring of v but do not correspond to fixpoints of the symmetry ΓZ A ω R (ρ). Given T f the forests (F (v)) v∈T f are i.i.d. and we let ζ denote a random variable that is distributed according to the size |F (v)|. Then its pgf is given by Let (T n , β n ) and T f n denote the tree (T , β) and fixpoint tree T f conditioned on the event |ΓZ A ω R (ρ)| = n (which is equivalent to |T | = n) and for any vertex v ∈ T f n let F n (v) and f n (v) denote the conditioned versions of F (v) and f (v). In the following lemma we show that w.h.p. each of the forests (F n (v)) v∈T f n has only O(log(n)) vertices and the size of the fixpoint-tree T f n concentrates around a constant multiple of n. Moreover, T f n / √ n converges weakly towards a constant multiple of the CRT T e with respect to the Gromov-Hausdorff metric.
Lemma 3.5. Suppose that ρ > 0 and that the function Then the following assertions hold: i) The offspring distribution ξ of the GWT T f and the random variable ζ have finite exponential moments. Moreover, ii) There are constants C > 0 and 0 < γ < 1 such that for all x ≥ 0 iii) For any vertex v let D v denote the d Yn -diameter of the subspace {v} ∪ F n (v) ⊂ Y n . Then there are constants C > 0 and 0 < γ < 1 such that for all h ≥ 0 iv) For any 0 < s < 1/2 it holds with high probability that .
v) We have that in the Gromov-Hausdorff sense as n ≡ 1 mod span(w) tends to infinity.
Proof. We start with the proof of claim i). By the assumption E(ρ + ,Ã ω R (ρ) + ) < ∞ it follows that the probability generating functions of ξ and ζ given in Equations (3.1) and (3.2) have radius of convergence strictly greater than 1. Hence ξ and ζ have finite exponential moments. The equations for the mean values and variance follow from the expressions of the pgfs. By Lemma 3.3 we have that E u (ρ,Ã ω R (ρ)) = 1. We proceed with claim ii). Let E denote the event that there is a vertex v ∈ T f with |F (v)| ≥ x. If |T | = n, then |T f | ≤ n. Using Lemma 3.3 it follows that P(E | |T | = n) = O(n 3/2 )P(E, |T | = n) = O(n 5/2 )γ x .
We now show claim iii). We may form a random metric space Y by constructing a metric d Y on the vertex set of ΓS(ρ) by patching together independent copies of the metrics δ R just as in the construction of the metric space Y n . Hence Y n is distributed like the space Y conditioned on having size n. For any vertex v of the fixpoint tree T f let D v denote the d Y -diameter of the subspace {v} ∪ F (v) ⊂ Y. Given h ≥ 0 let E denote the event that D v ≥ h for at least one vertex v ∈ T f . Given T f the family (F (v)) v∈T f is i.i.d., hence using Lemma 3.3 it follows that with o denoting the root of the fixpoint tree T f . By assumption, for any vertex u ∈ F (o), the distance d Y (o, u) is bounded by the sum of e d + T (e) many independent copies of a real-valued random variable χ ≥ 0 having finite exponential moments, with the sum index e ranging over all ancestors of the vertex u in the tree T . Clearly we have that with (χ i ) i∈N a family independent copies of χ. Moreover, P(|F (o)| = k) = P(ζ = k) = O(γ k 1 ) for some constant 0 < γ 1 < 1. By the deviation inequality given in Lemma 2.2 it follows that there are constants a, b > 0 such that γ k 1 P(χ 1 + . . . + χ k ≥ h/2) ≤ 2e −ak−bh for all k and h. Hence 2 ) for some constant 0 < γ 2 < 1. Hence Equality ( * ) implies that P(E | |Y| = n) = O(n 5/2 )γ h 2 and we are done.
We proceed with the proof of iv). Let 0 < s < 1/2 be given. Let E 1 n denote the event that This implies the event E 2 n that By definition of fixpoint tree and the forests F (v) we have that In particular, since claim i) implies that with high probability, it follows that w.h.p.
|T f n | ≥ n/ log(n) 2 . Thus we may bound the probability P(E 1 n | |T | = n) by Letting (ζ i ) i∈N denote a family of independent copies of the random variable ζ we may bound this further by with the sum index ranging over all integers in the interval from n/ log(n) 2 to n. By the deviation inequality in Lemma 2.2 this bound converges to zero as n tends to infinity. Thus w.h.p. |T f n | ∈ (1 ± n −s )n/(1 + E[ζ]). We may now deduce claim v). Set t := (1 + E[ζ])σ/2 with σ 2 = V[ξ]. Let g : K → R denote a Lipschitz-continous, bounded function defined on the space of isometry classes of compact metric spaces. Note that T f n conditioned on having size is distributed like T f conditioned on having size . Hence it is distributed like a ξ-Galton-Watson tree conditioned on having vertices, which we denote by T . It follows from claim ii) that with the sum index ranging over all integers ≡ 1 mod span(w) contained in the interval (1 ± n −s ) n 1+E [ζ] . Since g is Lipschitz-continuous, we have that for some constants a n, with sup (a n, ) → 0 as n ≡ 1 mod span(w) tends to infinity.  Our strategy for the proof of Theorem 1.3 will be to show that the Gromov-Hausdorff distance between the metric space Y n and a constant multiple of the fixpoint tree T f n converges in probability to zero as n becomes large. To this end, we are going to use the following decomposition introduced by Labelle [Lab81, Thm. A], in order to study the behavior of the R-structures along paths starting from the root in random enriched trees. The weighted species A ω R satisfies an isomorphism A ω R X · R κ (A ω R ). Here we assign weight 1 to each object of the species X . The pointing operator satisfies a product rule similar to the product rule for the derivative of smooth functions [Joy81]. Hence applying the pointing operator yields a weight-compatible isomorphism We may apply Joyal's implicit species theorem [Joy81, Th. 6] in order to unwind this recursion and obtain an isomorphism , α), v) in which the outer root v has height in the rooted tree A. The correspondence is illustrated in Figure 10. Again, this isomorphism is compatible with the weightings, and we may use it to construct the following sampler illustrated in Figure 11. Lemma 3.6. For any integer ≥ 0 and parameter x > 0 withÃ ( ) R (x) < ∞ consider the following recursive procedure ΓS ( ) (x) that samples a random Sym(R)-enriched plane tree (T ( ) , β ( ) ) together with a distinguished vertex r which we call the outer root. 1. If = 0 then return (an independent copy of ) the random enriched plane tree (T , β) from Lemma 3.4 with the outer root being the root-vertex of T . Otherwise, if ≥ 1, then proceed with the following steps.
2. Start with a root vertex o and draw a random R -symmetry (R, σ) from k≥0 Sym(R ) [k] with probability proportional to Set k := |R| and make a uniformly at random choice of a bijection f from the set Lemma 3.6 follows from a slight extension of the rules for the construction of Pólya-Boltzmann samplers [BFKV11,Ch. 5] to the setting of weighted species. The most important property of the above sampler is that σ)). We are now ready for the proof of Theorem 1.3. Recall that Y n is constructed by assigning metrics to the vertices of the random enriched treeÃ R n which we may draw by conditioning the sampler ΓZ A ω R (ρ) on having size n. In particular, we consider a coupling ofÃ R n with a Sym(R)-enriched tree (T n , β n ) given by the Sym(R)-enriched tree (T , β) of the sampler ΓS(ρ) conditioned on having size n. In this coupling, any vertex of Y n corresponds to the atom of an A R -symmetry whose fixpoints form the fixpoint tree T f n ⊂ T n and whose non-fixpoints are partitioned into small forests (F n (v)) v∈T f n which are attached to the fixpoints. Proof of Theorem 1.3. By Lemma 3.5 ii) it follows that with high probability all vertices v ∈ T f n have the property that the d Yn -diameter of the subspace {v} ∪ F (v) is at most O(log(n)). This implies that the Gromov-Hausdorff distance between the metric spaces (Y n , d Yn / √ n) and (T f n , d Yn / √ n) converges in probability to zero. Moreover, by Lemma 3.5 iii) we know that (T f n , cd T f n / √ n) with c = (1 + E[ζ])V[ξ]/2 converges weakly to the CRT T e . It remains to show that there is a constant c such that the Gromov-Hausdorff distance between (T f n , d Yn / √ n) and (T f n , c d T f n / √ n) converges in probability to zero. We define the random number η as follows. Choose a random R -symmetry (R, σ) from k≥0 Sym(R )[k] with probability proportional to its weight and let η denote the δ R -distance of the two distinct * -labels. Note that by our assumptions on the cycle index sum Z R κ we have that |R| has finite exponential moments. Moreover, the diameter of the metrci δ R is bounded by |R| many independent copies of a real-valued random variable χ ≥ 0 with finite exponential moments. Hence η has finite exponential moments. We are going to show that the Gromov-Hausdorff distance of (T f n , d Yn / √ n) and (T f n , E[η]d T f n / √ n) converges in probability to zero. By the discussion in the preceding paragraph this implies that ( −→ T e and we are done. Let s > 1 and t > 0 be arbitrary constants and set s n = log(n) s and t n = n t . Let > 0 be given and let E 1 denote the event that there exists a fixpoint v ∈ T f n and an ancestor u of v with the property that We are going to show that with high probability none of the events E 1 and E 2 takes place This suffices to show the claim: Take s = 2 and t = 1/4 and suppose that the complentary events E c 1 and E c 2 hold. Given vertices a = b in the tree T f n let x denote ther lowest common ancestor. If x ∈ {a, b} then we have By property E c 2 and the triangle inequality it follows that |R| = −R ≤ 2n 1/4 . Thus, regardless whether x ∈ {a, b}, it holds that  for a fixed constant C that does not depend on n or the points a and x. It follows that with D(T f n ) denoting the diameter. Thus holds with high probability. Since we may choose arbitrarily small, and D(T f n )/ √ n converges in distribution (to a multiple of the diameter of the CRT), it follows that d GH (Y n , E[η]T f n ) → 0 in probability and we are done.
For each finite subset U ∈ N and each R-structure R ∈ R[U ] let (δ i R ) i∈N 0 be a family of independent copies of the metric δ R . Given a A R -symmetry S = ((T, α), σ) with label set [k] for some k ≥ 0 we may form the family (δ S (v)) v∈T of random metrics by traversing bijectively the vertices of T in ascending order 1, 2, . . . k and assigning to each vertex v the "leftmost" unused copy from the list (δ 1 α(v) , δ 2 α(v) , . . .). The metrics can be patched together to a metric d S on the vertex set [k] of the tree T just as described in Section 1.2.
We may assume that all random variables considered so far are defined on the same probability space and that the metric d Yn of Y n coincides with the metric d Zn with Z n denoting the sampler ΓZ A ω R (ρ) conditioned on having size n. Given the family (δ i R ) R,i let H ⊂ ∞ k=0 Sym(A R )[k] denote the finite set symmetries of size n such that the event E 1 takes place if and only if Z n ∈ H. By the definition of the event E 1 for any symmetry S = ((T, α), σ) ∈ H we may choose a fixpoint v S of σ having the property that there exists an ancestor u in the tree T with Let S denote the height h T (v S ). Note that since v S is a fixpoint, the tupel (T, α, v S , σ) is a A ( S ) R -symmetry. By Lemma 3.3 the probability for the sampler ΓZ A ω R (ρ) to have size n is Θ(n −3/2 ) and we have that ρ R • A ω R (ρ) = 1. Hence Equation (3.3) implies that the conditional distribution of the event E 1 given (δ i R ) R,i equals -distance between spine vertices v i and v j is distributed like the sum η 1 + . . . + η |i−j| of independent copies (η i ) i of η. We know that η has finite exponential moments and hence by the deviation inequality in Lemma 2.2 the bound ( * ) converges to zero as n ≡ 1 mod span(w) tends to infinity. Thus with high probability E 1 does not hold. By the same arguments we may bound the probability for the event E 2 by Θ(n 3/2 ) n =1 min(sn, ) k=1 P(η 1 + . . . + η k ≥ t n ) which also converges to zero. This concludes the proof.
Proof of Lemma 1.4. It suffices to show that there are constants C, c, N > 0 such that for all n ≥ N and h ≥ √ n we have that P(H(X n ) ≥ h) ≤ C(exp(−ch 2 /n) + exp(−ch)).
For any fixpoint v ∈ T f n set (v) = u d + Tn (u) with the sum index u ranging over all ancestors of the vertex v in the plane tree T f n . Note that we are summing up the outdegrees in the tree T n and not in the tree T f n . Moreover, for any vertex y ∈ T n let v y denote its closest fixpoint, i.e. v y = y if y is a fixpoint and otherwise v y is the unique vertex with y ∈ F n (v y ). If y has height h Yn (y) ≥ 2h then h Yn (v y ) ≥ h or d Yn (u, v y ) ≥ h. Thus either H Yn (T f n ) ≥ h or there exists a fixpoint v ∈ T f n such that the d Yn -diameter D v of the subspace {v} ∪ F (v) is greater than or equal to h.
Let s > r > 0 be constants. Given h ≥ √ n let E 2h denote the event that H(X n ) ≥ h. It follows that E 2h ⊂ E h 0 ∪ E h 1 ∪ E h 2 ∪ E h 3 with the events E h i given as follows. E h 0 is the event that there exists a fixpoint v ∈ T f n with D v ≥ h. E h 1 is the event that H(T f n ) ≥ rh. E h 2 is the event that H(T f n ) ≤ rh and (v) ≥ sh for some fixpoint v ∈ T f n . E h 3 is the event that (v) ≤ sh for all fixpoints v ∈ T f n and H Yn (T f n ) ≥ h. We are going to show that if we choose r and s sufficiently small, then each of these events is sufficiently unlikely. By Lemma 3.5 we have that P(E h 0 ) = O(n 5/2 )γ h for some 0 < γ < 1. Hence there are constants C 0 , c 0 > 0 such that P(E h 0 ) ≤ C 0 exp(−c 0 h 2 /n) if h ≤ n and P(E h 0 ) ≤ C 0 exp(−c 0 h) if h ≥ n. In order to bound the probability for the event E h 1 note that the tree T f n conditioned on having size is distributed like T f conditioned on having size . That is, it is identically distributed to a ξ-Galton-Watson tree conditioned on having size which we denote by T . Hence P(|T f n | = )P(H(T ) ≥ rh).
By Inequality (2.1) there exist constants C 1 , c 1 > 0 that do not depend on n or h such that for all 1 ≤ ≤ n we have the tail bound P(H(T ) ≥ rh) ≤ C 1 exp(−c 1 r 2 h 2 / ) ≤ C 1 exp(−c 1 r 2 h 2 /n).
In particular, it holds that P(E h 1 ) ≤ C 1 exp(−c 1 r 2 h 2 /n) for all n and h. We proceed to bound the probability for the event E h 2 . Let Z n denote the sampler ΓZ A ω R (ρ) conditioned on having size n and let H ⊂ ∞ k=0 Sym(A R )[k] denote the set of A R -symmetries S = ((T, α), σ) having the property that the there exists a vertex v S in T with the property that S := h T (v S ) ≤ rh and u d + T (u) ≥ sh with the sum-index u ranging over all ancestors of the vertex v in the tree T . By Equation 3.3 we may bound the probability for the event E h 2 by P(E h 2 ) = P(Z n ∈ H) = O(n 3/2 ) S∈H P(ΓZ A ( S ) R (ρ) = (S, v S )).
By the deviation inequality in Lemma 2.2 it follows that there are constants c, λ > 0 such that the above quantity is bounded by a constant multiple of exp(5/2 log(n) + crh − λsh). We assumed that h ≥ √ n, hence if we choose r sufficiently small depending only on s, c and λ, it follows that there are constants C 2 , c 2 > 0 such that P(E h 2 ) ≤ C 2 exp(−c 2 h). It remains to treat the event E h 3 . By assumption, for any fixpoint v ∈ T f n we have that the height h Yn (v) is bounded by (v) many independent copies of a random variable χ having finite exponential moments. Thus

P(E h
3 ) ≤ nP(χ 1 + . . . + χ sh ≥ h) with (χ i ) i∈N a family of independent copies of χ. By the deviation inequality in Lemma 2.2 there are constants c, λ > 0 such that this quantity is bounded by a constant multiple of exp(log(n) + sh c − λh). We assumed that h ≥ √ n, hence we may bound this by exp(h(log(n)/ √ n + sc − λ)). If s is sufficiently small, then it follows that there are constants C 3 , c 3 > 0 such that P(E for all n and h ≥ √ n. This concludes the proof.