Asymptotics of heights in random trees constructed by aggregation

To each sequence $(a_n)$ of positive real numbers we associate a growing sequence $(T_n)$ of continuous trees built recursively by gluing at step $n$ a segment of length $a_n$ on a uniform point of the pre-existing tree, starting from a segment $T_1$ of length $a_1$. Previous works on that model focus on the influence of $(a_n)$ on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence $(a_n)$ is regularly varying with a non-negative index, so that the sequence $(T_n)$ exploses. We determine the asymptotics of the height of $T_n$ and of the subtrees of $T_n$ spanned by the root and $\ell$ points picked uniformly at random and independently in $T_n$, for all $\ell \in \mathbb N$.


Introduction
A well-known construction of the Brownian continuum random tree presented by Aldous in the first of his series of papers [2,3,4] works as follows. Consider a Poisson point process on R + with intensity tdt and "break" the half-line R + at each point of the process. This gives an ordered sequence of closed segments with random lengths. Take the first segment and glue on it the second segment at a point chosen uniformly at random (i.e. according to the normalized length measure). Then consider the continuous tree formed by these two first segments and glue on it the third segment at a point chosen uniformly at random. And so on. This gluing procedure, called the line-breaking construction by Aldous [2], gives in the limit a version of the Brownian CRT.
We are interested in a generalization of this construction, starting from any sequence of positive terms (a n , n ≥ 1). * Université Paris 13, Sorbonne Paris Cité LAGA, CNRS (UMR 7539) 93430 Villetaneuse, France. Email: haas@math.univ-paris13.fr For each n, we let b n denote a closed segment of length a n . The construction process then holds as above: we start with T 1 := b 1 and then recursively glue the segment b n on a point chosen uniformly on T n−1 , for all n ≥ 1. The trees T n are viewed as metric spaces, once endowed with their length metrics, which will be noted d in all cases. This yields in the limit a random real tree obtained as the completion of the increasing union of the trees T n , T := ∪ n≥1 T n that may be infinite. We let d denote its metric as well, and decide to root this tree at one of the two extremities of b 1 .
This model has been recently studied by Curien and Haas [10] and Amini et al. [5]. The paper [10] gives necessary conditions and sufficient conditions on the sequence (a n ) for T to be compact (equivalently bounded) and studies its Hausdorff dimension.
Typically, if a n ≤ n α+•(1) and a 1 + . . . + a n = n 1+α+•(1) for some α < 0, then almost surely the tree T is compact and its set of leaves has Hausdorff dimension 1/|α|, which ensures that the tree itself has Hausdorff dimension max(1, 1/|α|). This, as an example, retrieves the compactness of the Brownian CRT and that its Hausdorff dimension is 2. On the other hand, the tree T is almost surely unbounded as soon as the sequence (a n ) does not converge to 0. The issue of finding an exact condition on (a n ) for T to be bounded is still open. However, Amini et al. [5] obtained an exact condition for T to be bounded, provided that (a n ) is non-increasing. In that case, almost surely, Concerning related works, Ross and Wen [19] have recently shown that when the a n , n ≥ 1 are random lengths obtained by breaking the half-line with a Poisson point process with intensity t dt for some positive integer , the corresponding trees arise as scaling limits of some growing combinatorial trees constructed inhomogeneously, thus generalizing the convergence of trees constructed via Rémy's algorithm towards the Brownian CRT. There are also related constructions with different gluing rules.
Sénizergues [20] studies a generalization of the above (a n )-model where the segments are replaced by d-dimensional independent random metric measured spaces (d ∈ (0, ∞)) and the gluing rules depend both on the diameters and the measures of the metric spaces. He shows an unexpected and intriguing Hausdorff dimension. In other directions, Addario-Berry, Broutin and Goldschmidt [1] provide a line-breaking construction of the continuum limit of critical random graphs, extending Aldous' line-breaking construction to random real trees with vertex identifications; while Goldschmidt and Haas [14] propose a construction of the stable Lévy trees introduced by Duquesne, Le Gall and Le Jan [13,15] that generalizes Aldous' line-breaking construction to this class of trees (except in the Brownian case, the stable Lévy trees are not binary and the gluing procedure is then slightly more complex). See also [8], [12] and [18] for other related models.
The aim of the present paper is to examine the cases where the (a n )-model obviously leads to an unbounded tree and we will almost always assume that the sequence (a n ) is regularly varying with index α ≥ 0.
We recall that this means that for all c > 0, a cn a n −→ n→∞ c α , EJP 22 (2017), paper 21.
the prototype example being the sequence (n α ). We refer to Bingham et al. [7] for background on that topic. Our goal is to understand how the tree T n then grows as n → ∞. To that end, we will study the asymptotic behavior of the height of a typical point of T n and of the height of T n . We will see that in general these heights do not grow at the same rate. We will also complete the study of the height of a typical point first by providing a functional convergence, and second by studying the behavior of the subtrees of T n spanned by the root and points picked uniformly at random and independently in T n , for all ∈ N.
Height of a typical point and height of T n . We are interested in the asymptotic behavior of the following quantities: • D n : height of a typical point, i.e. given T n , we pick X n ∈ T n uniformly at random in T n and let D n = d(X n , root) be its distance to the root; • the height of the tree: In the particular case where all the lengths a n are identical, the sequence (T n ) can be coupled with a growing sequence of uniform recursive trees with i.i.d. uniform (0, 1) lengths on their edges. This is explained in Section 5. The asymptotic behavior of the height of a uniform vertex and the height of a random recursive tree without edge lengths (i.e. endowed with the graph distance) are well-known, [11,16,17]. From this and the strong law of large number, we immediately get the asymptotic of D n . The behavior of H n is less obvious. However, Broutin and Devroye [9] develop the material to study the height of random recursive trees with i.i.d. edge lengths, using the underlying branching structure and large deviation techniques. From this, we will deduce that: Theorem 1.1. If a n = 1 for all n ≥ 1, where β * is the unique solution in (0, ∞) to the equation 2(e β − 1) = βe β . Approximately, β * ∼ 1.594 and e β * /2β * ∼ 1.544.
This will be carried out in Section 5. Our main contribution concerns the cases where the index of regular variation α is strictly positive. In that case we introduce a random variable ξ (α) characterized by its Laplace The Lévy-Khintchine formula ensures that ξ (α) is infinitely divisible. Note also that ξ (α) is stochastically decreasing in α. Our main result is: Assume that (a n ) is regularly varying with index α > 0. Then, H n · ln(ln(n)) a n ln(n) a.s.
More precisely, in (i), E[exp(λa −1 n D n )] converges to E[exp(λξ (α) )] for all λ ∈ R, which in particular implies the convergence of all positive moments.
The proof of (i) is undertaken in Section 2.2 and relies on the powerful observation from [10] that D n can be written as the sum of i.i.d. random variables. The proof of (ii), and in particular of the lower bound, is more intricate. It relies on the second moment method and requires to get the joint distribution of the paths from the root to two points marked independently, uniformly in the tree T n (established in Section 3.1) as well as precise deviations bounds for the convergence (i) (established in Section 2.2). The core of the proof of (ii) is undertaken in Section 4.
The two previous statements on the asymptotic behavior of D n can actually be grouped together and slightly generalized as follows: Assume that (a n ) is regularly varying with index α ≥ 0. Then, where D ∞ denotes a positive random variable with finite expectation.
This will be explained in the remark around (2.8) in Section 2.2.
Height of the n-th leaf and height of a uniform leaf. In the recursive construction of (T n ), we can label the leaves L 1 , L 2 , . . . by order of appearance, so that the leaf L n belongs to the segment b n . We then let L n, denote a leaf chosen uniformly at random amongst the n leaves of T n . Theorem 1.2 (i) implies that when (a n ) varies regularly with index α > 0, and d(L n, , root) a n (d) where U is uniform on (0, 1), independent of ξ (α) . The first convergence is simply due to the fact that the distance d(L n , root) is distributed as a n + D n−1 , since the segment b n is inserted on a uniform point of T n−1 . The second convergence is explained in Section 2.2. When a n = 1 for all n, d(L n , root) and d(L n, , root) both divided by ln(n) converge to 1/2, almost surely and in probability respectively (see Section 5).
Functional convergence. The convergence of the height of a typical point can actually be improved into a functional convergence when the index of regular variation is strictly positive. As above, let X n be a point picked uniformly in T n and for each positive integer k ≤ n, let X n (k) denote its projection onto T k . Let then D n (k) := d (X n (k), root) , 1 ≤ k ≤ n be the non-decreasing sequence of distances of these branch-points to the root. If a climber decides to climb from the root to the typical point X n at speed 1, D n (k) is the time he will spend in T k . The proof of Theorem 1.2 (i) can be adapted to get the behavior as n → ∞ of the sequence (D n (k), 1 ≤ k ≤ n). To do so, introduce for α > 0 the càdlàg Markov process with independent, positive increments defined by (We note that ξ (α) (1) is distributed as the r.v. ξ (α) defined via (1.1).) This process is α-self-similar, in the sense that for all a > 0, for the Skorokhod topology on D([0, 1], R + ), the set of càdlàg functions from [0, 1] to R + . This is proved in Section 2.3.
Gromov-Prohorov-type convergence. Last, fix a positive integer, and given T n , let X (1) n , . . . , X ( ) n be points picked independently and uniformly at random in T n . Our goal is to describe the asymptotic behavior of T n ( ), the subtree of T n spanned by these marked points and the root. To that end, for all 1 ≤ i, j ≤ , we denote by B For regularly varying sequences of lengths (a n ), the tree T n ( ) appropriately rescaled converges to a "star-tree" with branches with random i.i.d. lengths. More precisely: (ii) Assume that (a n ) is regularly varying with index 0 and that Notation. Throughout the paper, we use the notation a i , for all n ∈ N.

Height of a typical point
Fix n, and given T n , let X n be a point picked uniformly on T n . The goal of this section is to establish different results on the distribution of the distance of this marked point to the root, mainly when the sequence (a n ) is regularly varying with a strictly positive index. Our approach entirely relies on the fact that this distance can be written as the sum of independent, non-negative random variables. More precisely, as noticed in [10], the distances D n (k) to the root of the projections of X n onto T k , k ≤ n can jointly be written in the following form: where U i , V i , 1 ≤ i ≤ n are all uniformly distributed on (0, 1) and independent. In particular, the distance D n of X n to the root writes To see this, we roughly proceed as follows. Consider the projection X n (n − 1) of X n onto T n−1 . By construction, it is uniformly distributed on T n−1 given T n−1 , and then: • either X n ∈ T n−1 and X n (n − 1) = X n , which occurs with probability A n−1 /A n , • or X n ∈ T n \T n−1 and d(X n , X n (n − 1)) = a n V n with V n uniform on (0, 1) and independent of T n−1 , which occurs with probability a n /A n .
Iterating this argument gives (2.1). An obvious consequence is that 3) The rest of this section is organized as follows. In Section 2.1 we start by recalling some classical bounds for regularly varying sequences that will be used throughout the paper.
The first part of Section 2.2 concerns the asymptotic behavior of the height D n , with the proofs of Theorem 1.2 (i) and its corollaries (1.2), as well as Proposition 1.3. The second part of Section 2.2 is devoted to the implementation of bounds (Lemma 2.3) that will be crucial for the proof of Theorem 1.2 (ii) on the behavior of the height of T n , proof that will be undertaken in Section 4. Last, Section 2.3 contains the proof of Proposition 1.4.

Bounds for regularly varying sequences
Assume that (a n ) is regularly varying with index α ≥ 0. We recall some classical bounds that will be useful at different places in the paper.

One dimensional convergence and deviations
For α > 0, recall the definition of the random variable ξ (α) defined via its Laplace with φ (α) given by (1.1). With the expression (2.2), it is easy to find the asymptotic behavior of (D n ) by computing its Laplace transform and then get Theorem 1.2 (i). We more precisely have: EJP 22 (2017), paper 21. Lemma 2.1. Assume that (a n ) is regularly varying with index α > 0. Then, (ii) For all c > 1, there exists n c such that for all n ≥ n c and all λ ≥ 0, where U, V are uniform on (0, 1) and independent (if a i = 0 for some i we use the convention (exp(0) − 1)/0 = 1). Now assume that λ > 0 (the following lines hold similarly for λ < 0 by adapting the bounds). Using (2.4), (2.5) together with the fact that ln(1 + x) ∼ x as x → 0 and that x → x −1 (exp(x) − 1) is increasing on (0, ∞) and converges to 1 as x → 0, leads to the existence of an integer j ε such that for n ≥ j ε we recognize Riemann sums in the lower and upper bounds, which, letting first n ↑ ∞ and then ε ↓ 0 gives It is easy to see with the change of variables y = x α in the integral and then the power series expansion of the exponential function that this expression of φ (α) (λ) indeed corresponds to (1.1).
(ii). Fix c > 1. Using the upper bound (2.7) and the fact that we see that for all 0 < η < α and then for all n large enough EJP 22 (2017), paper 21.
which, clearly, is smaller than ηλ exp(ηλ) for n large enough and all λ ≥ 0. Gathering this together, we get that for all n large enough (depending on η) and all λ ≥ 0, .
Remark (height of a uniform leaf). We keep the notation of the introduction and let L n, denote a leaf chosen uniformly at random amongst the n leaves of T n . Then the previous result implies that when (a n ) is regularly varying with index α > 0, with U uniformly distributed on (0, 1) and independent of ξ (α) . To see this, one could use that the distribution of (1 + ξ (α) )U α is characterized by its positive moments (since it has exponential moments, since ξ (α) has), together with the fact that for each p ≥ 0, Since d(L i , root) − a i is uniformly distributed on T i−1 (by construction) we know from the previous lemma that, divided by a i , it converges in distribution to ξ (α) , and that more precisely there is convergence of all positive and exponential moments. Together with (2.4), this leads to the convergence of E[(d(L n, , root)/a n ) p ] to E[(1 + ξ (α) ) p ]/(αp + 1), as expected.
Remark (other sequences (a n )). It is easy to adapt Part (i) of the proof to get that for a general sequence (a n ) of positive terms such that ( It is easy to check that the above condition on (a n ) holds if (a n ) is regularly varying with index 0 and In particular this recovers the first part of Theorem 1.1. To illustrate with other 0regularly varying sequences, consider a n = (ln(n)) γ , γ ∈ R. Then: where the last line is due to the fact that (D n ) is stochastically increasing (by (2.2)) and that lim n E[D n ] is finite when γ < −1 (by (2.3),(2.5)), which implies that (D n ) converges in distribution to a r.v. D ∞ with finite expectation. Note that this last argument actually holds for any sequence (a n ) such that is regularly varying, necessarily with index 0). All these remarks lead to Proposition 1.3, using again (2.6) when α > 0.
We come back to the case where α > 0 and note the following behavior of the maximum of n i.i.d. copies of ξ (α) . Proof for all u > 0. With the above estimates, it is straightforward that the right-hand side converges to 0 when u > 1 and to −∞ when u < 1.
We will not directly use this result later in the paper, but this may be seen as a hint that the height H n may be asymptotically proportional to n α ln(n)/ ln(ln(n)). To prove this rigorously, we will actually use the following estimates.
Proof. Of course, since D n − D n ( nc ) ≤ D n , we only need to prove the convergence to 0 in (i) for γ < γ and the convergence to ∞ in (ii) for γ > γ.

Functional convergence
In this section we prove Proposition 1.4. To lighten notation, we let for all n ∈ N ξ n (t) : uniform on (0, 1) (recall the construction (2.1)). Our goal is to prove that the process (ξ n ) converges to the process ξ (α) defined by (1.3) for the Skorokhod topology on D([0, 1], R + ). We start by proving the finite-dimensional convergence, relying on manipulations done in Section 2.2. Then we use Aldous' tightness criterion to conclude that the convergence holds with respect to the topology of Skorokhod.
Finite-dimensional convergence. The processes ξ n , n ≥ 1 and ξ (α) all have independent increments, by construction. It remains to prove that On the other hand, Campbell's theorem applied to the Poisson point process EJP 22 (2017), paper 21.
which indeed coincides with the above limit of E exp λ ξ n (t) − ξ n (s) .
where S n is the set of stopping times with respect to the filtration generated by the process ξ n .
The first point is obvious, since the processes ξ n are non-decreasing and we already know that ξ n (1) converges in distribution. For the second point, note that if τ ∈ S n , then nτ is a stopping time with respect to the filtration generated by the process We may assume that ε < α. Then, using P(X > ε) ≤ ε −1 E[X] for any non-negative r.v. X, we get where C α,ε depends only on α, ε. To get the last line we have used that either α −ε−1 ≥ 0 and then (since k + 1 ≤ 2n) where the last inequality is obtained by considering the first term in the minimum when k + 1 ≤ nδ and the second term when k + 1 > nδ.

Multiple marking
In order to prove Theorem 1.2 (ii), we need the joint distribution of the paths from the root to two points marked independently, uniformly in the tree T n . This is studied in Section 3.1. Then in Section 3.2, we turn to marked points and the proof of Proposition 1.5.

Marking two points
The result of this section is available for any sequence (a n ) of positive terms. Given T n , let X n , X n denote two points taken independently and uniformly in T n , and D n (k) , 1 ≤ k ≤ n -we recall that the marginals are given by (2.1). To that end, we introduce a sequence B (i,1) , B (i,2) , i ≥ 1 of independent pairs of random variables defined by: Note the two following facts (which will be useful later on): • B (i,1) (resp. B (i,2) ) is stochastically smaller than a Bernoulli r.v. with success parameter a i /A i • the distribution of B (i,1) given that B (i,2) = 0 (resp. B (i,2) given that B (i,1) = 0) is a Bernoulli r.v. with success parameter a i /A i .
i , i ≥ 1 be independent r.v. uniformly distributed on (0, 1), all independent of a sequence ((B (i,1) , B (i,2) ), i ≥ 1) of independent pairs of Bernoulli r.v. distributed as (3.1). Then for all n ≥ 1 and all bounded continuous functions f : R 2×n → R, This lemma implies in particular that the distribution of the splitting index S n (2) of the two paths linking respectively X (1) n and X (2) n to the root, i.e.
where p k (X (i) n ), i = 1, 2 denotes the projection of X (i) n onto T k , is given by (which is indeed a probability distribution!). Moreover, given S n (2) = κ, the dependence of the two paths above the index κ + 1 is only driven by pairs of random variables B (i,1) , B (i,2) , i ≥ κ + 1, as described in (3.3).
Proof. We proceed by induction on n ≥ 1. For n = 1, the formula of the lemma reduces which is obviously true since the two marked points are independently and uniformly distributed on a segment of length a 1 . Consider now an integer n ≥ 2 and assume that the formula of the lemma holds for n − 1. When marking X (1) n , X n , four disjoint situations may arise: • with probability (a n /A n ) 2 , the two marked points are on the branch b n . Conditionally on this event, D n (k), 1 ≤ k ≤ n − 1 which corresponds to the path to the root of a point uniformly distributed on T n−1 , which is distributed as  n (k), k ≤ n − 1), and uniformly distributed on b n , which has length a n . All this leads to the term κ = n in the sum (3.2). • with probability A n−1 a n /A 2 n , X where we have used for the second equality that A n−1 a n A 2 n = 1 − a n A n 2 × P B (n,1) = 0, B (n,2) = 1 .
• with probability A n−1 a n /A 2 n , X n ∈ T n−1 and X (1) n ∈ b n , which is symmetric to the previous case. n (n − 1) and D (1) n (k), 1 ≤ k ≤ n − 1 and D (2) n (k), 1 ≤ k ≤ n − 1 correspond to the paths to the root of two points marked independently, uniformly in T n−1 . Their joint distribution is therefore given by the induction hypothesis, and setting for κ ≤ n − 1, ∆ where we have used for the second equality that Gathering these four situations finally leads to the formula of the lemma for n.

Marking points and behavior of T n ( )
The goal of this section is to prove Proposition 1.5. We start with a few notation.
For each n, given T n , let X In the tree T n ( ), the subtree of T n spanned from the root and X (1) n , . . . , X ( ) n , we let, using the notation of the introduction, be the point amongst the B (i,j) n , 1 ≤ i = j ≤ the farthest from the root (note that it is well-defined a.s.). We may and will also see B n ( ) as a point of T n .
We will need the following random variables.  In order to study the asymptotic behavior of (T n ( )), we set up the following lemma, which is similar to Lemma 3.1, although less explicit.
where the random variables V (j) ,1) , . . . , B (i, ) are distributed via (3.5), ∀i ≥ 1, independently of each other and independently of (V (j) Proof. The proof is similar to that of Lemma 3.1 and holds by induction on n > k. We sketch it briefly. For n = k + 1 the statement is obvious since both −uplets are then equal to (0, . . . , 0). Assume now that the statement holds for some n > k. Then observe what happens for n + 1: given that B n+1 ( ) ∈ T k , two situations may occur: • either none of the marked points belongs to the segment b n+1 . This occurs with a probability proportional to (A n ) and then • or a unique marked point belongs to the segment b n+1 . The probability that X where V is uniform on (0, 1) and independent of D This leads to the statement for n + 1.
Proof of Proposition 1.5. Throughout this proof it is assumed that (a n ) is regularly varying with index α > 0 (the proof is identical under the assumptions (ii) of Proposition 1.5).
With the notation of this section, our goal is to prove that since a n → ∞ and d(B n ( ), root) ≤ 1≤i =j≤ d(B (i,j) n , root), which is stochastically bounded since the splitting index S n (2) of the paths of two marked points converges in distribution, by (3.4). By Slutsky's Theorem, it remains to prove that We start by observing that for all k ≥ 1, which obviously leads to (since a n → ∞) The above observation relies on the following consequence of Lemma 3.2: for all (λ i ) 1≤i≤ ∈ R and all n > k, n , 1 ≤ i ≤ ) and f : R → R be a continuous, bounded function. Fix ε > 0. There exists k ε ∈ N such that P(B n ( ) / ∈ T kε ) ≤ ε for all n, since, as already mentioned, the splitting index of the paths of two marked points converges in distribution, by (3.4). Then, writing Letting ε → 0 gives the result.
4 Height of T n when α > 0 Throughout this section we assume that (a n ) is regularly varying with index α > 0.
Our goal is to prove that H n · ln(ln(n)) a n ln(n) Asymptotics of heights in random trees constructed by aggregation (Theorem 1.2 (ii)). We split the proof into two parts, starting with the fact that lim sup n→∞ H n · ln(ln(n)) a n ln(n) ≤ 1 a.s., (4.1) which is an easy consequence of Borel-Cantelli's lemma and Lemma 2.3 (i). We will then show that lim inf n→∞ H n · ln(ln(n)) a n ln(n) ≥ 1 a.s., (4.2) using the second moment method and, again, Borel-Cantelli's lemma. To carry this out, we will use Lemma 3.1 on the two marked points, as well as the estimates of Lemma 2.3 (ii).

Proof of the limsup (4.1)
In the infinite tree ∪ n≥1 T n , label the leaves by order of appearance: for each i ≥ 1, the leaf L i is the one that belongs to the branch b i . Then consider for i ≥ 2 the projection of L i onto T i−1 and denote by D i−1 the distance of this projection to the root, which is distributed as D i−1 . Let D 0 = 0 and note that Hence by Borel-Cantelli's lemma, almost surely for all i large enough. This leads, together with the fact that (a i ) is regularly varyingsee in particular (2.4) -to the almost sure existence of a (random) i 0 such that D i−1 + a i < c 1 a n ln(n) ln(ln(n)) for all n ≥ i ≥ i 0 . Hence, lim sup n→∞ H n · ln(ln(n)) a n ln(n) ≤ c 1 a.s.

Proof of the liminf (4.2)
Let X (i) n , 1 ≤ i ≤ n be n points marked independently and uniformly in T n . Then let D (i) n , 1 ≤ i ≤ n denote their respective distances to the root, and for all k < n, D (i) n (k), 1 ≤ i ≤ n denote the distances to the root of their respective projections onto T k . Of course, H n ≥ max 1≤i≤n D (i) n and it is sufficient to prove the liminf for this maximum of dependent random variables. To that end, we first establish the following lemma, using the second moment method.

(4.3)
Since H n is larger than the maximum involved in this probability, this immediately implies that P H n · ln(ln(n)) a n ln(n) ≤ γ −→ n→∞ 0 for all γ < 1.
This is however not sufficient since we want an almost sure bound for the liminf (4.2).
We will turn to this conclusion later on. We first prove the lemma.
Proof of Lemma 4.1. We start with standard arguments, in order to use the second moment method. Fix γ ∈ (0, 1) and introduce Since the sequence (D Note that with this notation, (4.3) rewrites P (S n = 0) ≤ n γ−1+• (1) . To prove this upper bound, we use the second moment method:

By Lemma 2.3 (ii), we know that nP A
(1) n = n 1−γ+• (1) . It remains to show that To that end, recall the notation and statement of Lemma 3.1: κ ( nc ) a n > γ ln(n) ln(ln(n)) , j = 1, 2 where p κ := aκ (i) First, using the notation of Section 3.1 and the remarks just before Lemma 3.1, we see that κ ( nc ) a n > γ ln(n) ln(ln(n)) The first inequality is due to the fact that the sum conditional on B (i,1) = 0 is a Bernoulli r.v. with success parameter a i /A i , and moreover B (i,2) = 0 a.s. when B (i,1) = 1. The second inequality follows immediately from Lemma 3.1.

(4.4)
Now let c ∈ (0, 1) and note that with H nc and max 1≤i≤n D H n a n ≤ γ ln(n) ln(ln(n)) ≤ P H nc a n ≤ γ ln(n) ln(ln(n)) by (4.4) applied to nc instead of n (together with the regular variation assumption on (a n )) and Lemma 4.1. Next, fix an integer k such that (1 − γ)k > 1. Iterating the previous argument, we get that P H n a n ≤ γ ln(n) ln(ln(n)) ≤ n γ k−1 j=0 c −αj −k+• (1) .
We now choose c ∈ (0, 1) sufficiently close to 1 so that γ k−1 j=0 c −αj −k < −1 and conclude with Borel-Cantelli lemma that almost surely H n · ln(ln(n)) a n ln(n) > γ for all n large enough.
5 The case a n = 1 The goal of this section is to prove Theorem 1.1. To that end we start by associating to a sequence (T n ) built recursively from a sequence (a n ) of positive lengths (with no constraints on the a n s for the moment) a sequence of graph-theoretic trees (R n ) that codes its genealogy as follows: • R 1 is the tree composed by a unique vertex, labeled 1 • if in T n the branch b n is glued on the branch b i , i < n, then R n is obtained from R n−1 by grafting a new vertex, labeled n , to the vertex i .
The vertex 1 is considered as the root of R n , ∀n ≥ 1. This sequence of genealogical trees has been used by [5] to study the boundedness of ∪ n≥1 T n . From now on it is assumed that a n = 1 for all n ≥ 1. In that case, for all n, R n is obtained by grafting the new vertex n to one vertex chosen uniformly at random amongst the n − 1 vertices of R n−1 . Hence R n is a uniform recursive tree with n leaves. Let d Rn denote the graph distance on R n . It is well-known that  U7 U4 Figure 1: On the left, a version of the tree T 8 . On the right, the associated genealogical tree with edge-lengths R 8 . The discrete (graph-theoretic) tree R 8 is obtained from R 8 by forgetting the uniform lengths U i , 2 ≤ i ≤ 8.
see [11,16,17]. Next we add lengths to the edges of the trees R n , n ≥ 2. By construction, there exists a sequence of i.i.d. uniform r.v. U i , i ≥ 1 such that in ∪ n≥1 T n , where the sequence b 1 → b i1 → . . . → b i k → b i represents the segments involved in the path from the root to L i (recall from the introduction that the leaves are labelled by order of insertion). For all n and all 2 ≤ i ≤ n, we decide to allocate the length U i to the edge in R n between the vertex i and its parent. We denote by R n this new tree with edge-lengths and by d Rn the corresponding metric, so that finally, d(L i , root) = d Rn ( i , 1 ) + 1, for all leaves L i ∈ T n . Height of a typical vertex in T n , height of leaf L n , height of a uniform leaf of T n .
The strong law of large numbers and the convergence on the left of (5.1) then clearly yield that d Rn ( n , 1 )/ ln(n) converges a.s. to 1/2. This in turn yields that d(L n , root)/ ln(n) converges a.s. to 1/2 and that D n ln(n) P −→ n→∞ 1 2 since b n+1 is inserted on a uniform point of T n . (More precisely, if we note, for each n, D n the distance to the root of the insertion point of b n+1 on T n , we obtain versions of the D n s that converge almost surely: D n / ln(n) → 1/2 a.s.).
Moreover, from the (a.s.) convergence of d(L n , root)/ ln(n) to 1/2, it is easy to get the convergence in probability of d(L n, , root)/ ln(n) to 1/2, where L n, is a uniform leaf of T n . We let the reader adapt the proof seen in Section 2.2 for regularly varying sequences (a n ) with a strictly positive index.