Local limits of conditioned Galton-Watson trees II: the condensation case

We provide a complete picture of the local convergence of critical or subcritical Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set. The generic case, where the limit is a random tree with an infinite spine has been treated in a previous paper. We focus here on the non-generic case, where the limit is a random tree with a node with infinite out-degree. This case corresponds to the so-called condensation phenomenon.

Let µ(p) = +∞ n=0 np(n) be its mean. If µ(p) < 1 (resp. µ(p) = 1, µ(p) > 1), we say that the offspring distribution and the associated GW tree are sub-critical (resp. critical, supercritical). In the critical and sub-critical cases, the tree is a.s. finite, but Kesten considered in [10] the limit of a sub-critical or critical tree conditioned to have height greater than n. When n goes to infinity, this conditioned tree converges in distribution to the so-called size-biased GW tree. This random tree has an infinite spine on which are grafted a random number of independent GW trees with the same offspring distribution p. This limit tree can be seen as the GW tree conditioned on non-extinction.
Since then, other conditionings have been considered for critical GW trees: large total progeny see Kennedy [9] and Geiger and Kaufmann [5], large number of leaves see Curien and Kortchemski [3]. In [1], we generalized those previous results by conditioning the GW tree to have a large number of individuals whose number of offspring belongs to a set A ⊂ N. Let (2) p(A) = k∈A p(k).
If p(A) > 0, then the limiting tree is again the same size-biased tree as for Kesten [10].
However, the results are different in the subcritical case. We first define for an offspring distribution p that satisfies (1) and a set A ⊂ N such that p(A) > 0 a modified offspring distribution p A,θ by: where the normalizing constant c A (θ) is given by: where X is a random variable distributed according to p. Let I A be the set of positive θ for which p A,θ is a probability distribution. If p is sub-critical, according to Lemma 5.2, either there exists (a unique) θ c A ∈ I A such that p A,θ c A is critical or θ * A := max I A ∈ I A and p A,θ * A is sub-critical. We shall say, see Definition 5.3, that p is generic for the set A in the former case and that p is non-generic for the set A in the latter case. See Lemma 5.4 and Remark 5.5 on the non-generic property.
For a tree t, let L A (t) be the set of nodes of t whose number of offspring belongs to A and L A (t) be its cardinal (see definition in Section 6). It is proven in [1] that, for every θ ∈ I A , if τ is a GW tree with offspring distribution p and τ A,θ is a GW tree with offspring distribution p A,θ , then the conditional distributions of τ given {L A (τ ) = n} and that of τ A,θ given {L A (τ A,θ ) = n} are the same. Therefore, if p is generic for the set A, that is there exists a θ c A ∈ I A such that p θ c A ,A is critical, then the GW tree τ conditioned on L A (τ ) being large converges to the size-biased tree associated with p A,θ c A . When the sub-critical offspring distribution is non-generic for N, a condensation phenomenon has been observed when conditioning with respect to the total population size, see Jonnsson and Stefansson [7] and Janson [6]: the limiting tree is no more the size-biased tree but a tree that contains a single node with infinitely many offspring. The goal of this paper is to give a short proof of this result and to show that such a condensation also appears when p is non-generic for A and conditioning by L A (τ ) being large. This and [1] give a complete description of the limit in distribution of a critical or subcritical GW tree τ conditioned on {L A (τ ) = n} as n goes to infinity.
We summarize this complete description as follows. Let p be an offspring distribution that satisfies (1) which is critical or sub-critical (that is µ(p) ≤ 1). Let τ * (p) denote the random tree which is defined by: i) There are two types of nodes: normal and special.
ii) The root is special. iii) Normal nodes have offspring distribution p. iv) Special nodes have offspring distribution the biased distributionp on N ∪ {+∞} defined by:p v) The offsprings of all the nodes are independent of each others. vi) All the children of a normal node are normal. vii) When a special node gets a finite number of children, one of them is selected uniformly at random and is special while the others are normal. viii) When a special node gets an infinite number of children, all of them are normal. Notice that: • If p is critical, then a.s. τ * (p) has one infinite spine and all its nodes have finite degrees. This is the size-biased tree considered in [10]. • If µ(p) < 1 then a.s. τ * (p) has exactly one node of infinite degree and no infinite spine. This tree has been considered in [7,6].
We define p * A as: critical case (µ(p) = 1): subcritical and generic for A (µ(p) < 1 and there exists (a unique) θ c subcritical and non-generic for A (µ(p) < 1 and µ(p A,θ * A ) < 1): We state our main result (the convergence of random discrete trees is precisely defined in Section 2 and GW trees are presented in Section 3). Theorem 1.2. Let τ be a GW tree with offspring distribution p which satisfies (1) and µ(p) ≤ 1. Let A ⊂ N such that p(A) > 0. We have the following convergence in distribution: where the limit is understood along the infinite subsequence {n ∈ N * ; P(L A (τ ) = n) > 0}, as well as: The theorem has already been proven in the critical case and the subcritical generic case in [1]. We concentrate here on the case of the subcritical non-generic case. The non-generic case for A = N, 0 ∈ A, 0 ∈ A are respectively proven in Sections 4, 6 and 7. Let us add that a subcritical offspring distribution p is either generic for all A ⊂ N such that p(A) > 0 or non-generic at least for {0} and eventually for other sets and generic for other sets A such that p(A) > 0, see Lemma 5.4. It is not possible for a subcritical offspring distribution p to be non-generic for all A ⊂ N such that p(A) > 0, see Remark 5.5. By considering the last example of Remark 5.5, we exhibit a distribution p which is non-generic for {0} but generic for N. Thus the associated GW tree conditioned on having n vertices converges in distribution (as n goes to infinity) to a tree with an infinite spine whereas the same tree conditioned on having n leaves converges in distribution to a tree with an infinite node.
In Section 2, we recall the setting of the discrete trees (which is close to [1], but has to include discrete trees with infinite nodes). We also give in Lemma 2.2, in the same spirit of Lemma 2.1 in [1], a convergence determining class which is the key result to prove the convergence in the non-generic case. Section 3 is devoted to some remarks on GW trees. We study in detail the distribution p A,θ defined by (3) in Section 5. The proof of Theorem 1.2 is given in the following three sections. More precisely, the case A = N is presented in Section 4. This provides an elementary and self-contained proof of the results from [7,6]. The case 0 ∈ A can be handled in the same spirit, see Section 6, using that the set L A (τ ) can be encoded into a GW tree τ A , see [11] or [14]. Notice that if 0 ∈ A, then L A (τ ), when non empty, can also be encoded into a GW tree τ A , see [14]. However, we didn't use this result, but rather use in Section 7 a more technical version of the previous proofs to treat the case 0 ∈ A. We prove in the appendix, Section 8, consequences of the strong ratio limit property we used in the previous sections.

The set of discrete trees
We recall Neveu's formalism [13] for ordered rooted trees. We set U = n≥0 (N * ) n the set of finite sequences of positive integers with the convention (N * ) 0 = {∅}. For n ≥ 0 and u = (u 1 , . . . , u n ) ∈ U , we set |u| = n the length of u and: with the convention |∅| = |∅| ∞ = 0. We will call |u| ∞ the norm of u although it is not a norm since U is not even a vector space. If u and v are two sequences of U , we denote by uv the concatenation of the two sequences, with the convention that uv = u if v = ∅ and uv = v if u = ∅. The set of ancestors of u is the set: The most recent common ancestor of a subset s of U , denoted by MRCA(s), is the unique A tree t is a subset of U that satisfies: • For every u ∈ t, there exists k u (t) ∈ N ∪ {+∞} such that, for every positive integer i, ui ∈ t iff 1 ≤ i ≤ k u (t). The integer k u (t) represents the number of offsprings of the vertex u ∈ t. (Notice that k u (t) has to be finite in [1], whereas k u (t) might take the value +∞ here.) The vertex u ∈ t is called a leaf if k u (t) = 0 and it is said infinite if k u (t) = +∞. By convention, we shall set k u (t) = −1 if u ∈ t. The vertex ∅ is called the root of t. We set: Let t be a tree. The set of its leaves is L 0 (t) = {u ∈ t; k u (t) = 0}. Its height and its "norm" are resp. defined by H(t) = sup{|u|, u ∈ t} and H ∞ (t) = sup{|u| ∞ , u ∈ t} = max(H(t), sup{k u (t), u ∈ t}); they can be infinite. For u ∈ t, we define the sub-tree S u (t) of t "above" u as: For u ∈ t \ L 0 (t), we also define the forest F u (t) "above" u as the following sequence of trees: For u ∈ t \ {∅}, we also define the sub-tree S u (t) of t "below" u as: Notice that u ∈ S u (t).
For v = (v k , k ∈ N * ) ∈ (N * ) N , we setv n = (v 1 , . . . , v n ) for n ∈ N, with the convention thatv 0 = ∅ andv = {v n , n ∈ N} defines a tree consisting of an infinite spine or branch. We denote by T ∞ the set of trees. We denote by T 0 the subset of finite trees, ∞ the subset of trees with norm less than h, by T * 0 the subset of trees with no infinite branch, T * 0 = {t ∈ T ∞ ; ∀v ∈ (N * ) N ,v ⊂ t}, and by T 2 the subset of trees with no infinite branch and with exactly one infinite vertex, For h ∈ N, the restriction function r h,∞ from T ∞ to T ∞ is defined by: We endow the set T ∞ with the ultra-metric distance A sequence (t n , n ∈ N) of trees converges to a tree t with respect to the distance d ∞ if and only if, for every h ∈ N, for n large enough, that is for all u ∈ U , lim n→+∞ k u (t n ) = k u (t) ∈ N ∪ {−1, +∞}. The Borel σ-field associated with the distance d ∞ is the smallest σ-field containing the singletons for which the restrictions functions (r h,∞ , h ∈ N) are measurable. With this distance, the restriction functions are contractant. Since T 0 is dense in T ∞ and (T ∞ , d ∞ ) is complete and compact, we get that (T ∞ , d ∞ ) is a compact Polish metric space.
Remark 2.1. In [1], we considered the subset of trees with no infinite vertex. On T, we defined the distance: with r h (t) = {u ∈ t, |u| ≤ h}. Notice that (T, d) is Polish but not compact and that T is not closed in (T ∞ , d ∞ ). If a sequence (t n , n ∈ N * ) converges in (T, d) then it converges in (T ∞ , d ∞ ). And if a sequence (t n , n ∈ N * ) of elements of T converges in (T ∞ , d ∞ ) to a limit in T then it converges to the same limit in (T, d).

Consider the closed ball
for some t ∈ T ∞ and h ∈ N and notice that: Since the distance is ultra-metric, the closed balls are open and the open balls are closed, and the intersection of two balls is either empty or one of them. We deduce that the family is a π-system, and Theorem 2.3 in [2] implies that this family is convergence determining for the convergence in distribution. Let (T n , n ∈ N * ) and T be T ∞ -valued random variables. We denote by dist (T ) the distribution of the random variable T (which is uniquely determined by the sequence of distributions of r h,∞ (T ) for every h ≥ 0), and we denote: for the convergence in distribution of the sequence (T n , n ∈ N * ) to T . Notice that this convergence in distribution is equivalent to the finite dimensional convergence in distribution of (k u (T n ), u ∈ U ) to (k u (T ), u ∈ U ) as n goes to infinity. We deduce from the portmanteau theorem that the sequence (T n , n ∈ N * ) converges in distribution to T if and only if for all h ∈ N, t ∈ T (h) As we shall only consider T 0 -valued random variables that converge in distribution to a T 2 -valued random variable, we give an other characterization of convergence in distribution that holds for this restriction. To present this result, we introduce some notations. If v = (v 1 , . . . , v n ) ∈ U , with n > 0, and k ∈ N, we define the shift of v by k as θ(v, k) the tree obtained by grafting the tree s at x on "the right" of the tree t, with the convention that t ⊛ (s, x) = t if s = {∅} is the tree reduced to its root. Notice that if x is a leaf of t and s ∈ T, then this definition coincides with the one given in [1].
For every t ∈ T 0 and every x ∈ t, we consider the set of trees obtained by grafting a tree at x on "the right" of t: as well as for k ∈ N: the subsets of T(t, x) such that the number of offspring of x are resp. k and k or more. It is easy to see that T + (t, x, k) is closed. It is also open, as for all s ∈ T + (t, x, k) we have that Moreover, notice that the set T 2 is a Borel subset of the set T. The next lemma gives another criterion for the convergence in distribution in T 0 ∪ T 2 . Its proof is very similar to the proof of Lemma 2.1 in [1].
Lemma 2.2. Let (T n , n ∈ N * ) and T be T ∞ -valued random variables which belong a.s. to T 0 ∪ T 2 . The sequence (T n , n ∈ N * ) converges in distribution to T if and only if for every t ∈ T 0 , x ∈ t and k ∈ N, we have: be the subset of trees with only one infinite spine (or branch). We give in [1] a characterization of the convergence in T 0 ∪T 1 as follows. Let (T n , n ∈ N * ) and T be T-valued random variables which belong a.s. to T 0 ∪ T 1 . The sequence (T n , n ∈ N * ) converges in distribution to T if and only if (9) holds for every t ∈ T 0 , x ∈ L 0 (t) and k = 0. In a sense, the convergence in T 0 ∪ T 1 is thus easier to check.
Borel sets on T 0 T 2 forms a π-system since we have in the other (non-symmetric) cases.
For every h ∈ N and every t ∈ T In particular F is a separating class in T 0 ∪ T 2 . Since A ∈ F is closed and open as well, according to Theorem 2.3 of [2], to prove that the family F is a convergence determining class, it is enough to check that, for all t ∈ T 0 ∪ T 2 and h ∈ N, there exists A ∈ F such that: Since we can find such a s and x such that |x| ∞ is arbitrary large, we deduce that (10) is satisfied. This proves that the family F is a convergence determining class in T 0 ∪ T 2 . Since, for t ∈ T 0 , x ∈ t and k ∈ N, the sets T + (t, x, k) and {t} are open and closed, we deduce from the portmanteau theorem that if (T n , n ∈ N * ) converges in distribution to T , then (9) holds for every t ∈ T 0 , x ∈ t and k ∈ N.

GW trees
3.1. Definition. Let p = (p(n), n ∈ N) be a probability distribution on the set of the nonnegative integers. We assume that p satisfies (1). Let g(z) = k∈N p(k) z k be the generating function of p. We denote by ρ(p) its convergence radius and we will write ρ for ρ(p) when it is clear from the context. We say that p is aperiodic if {k; p(k) > 0} ⊂ dN implies d = 1.
A T-valued random variable τ is a Galton-Watson (GW) tree with offspring distribution p if the distribution of k ∅ (τ ) is p and for n ∈ N * , conditionally on {k ∅ (τ ) = n}, the sub-trees (S 1 (τ ), S 2 (τ ), . . . , S n (τ )) are independent and distributed as the original tree τ . Equivalently, for every h ∈ N * and t ∈ T (h) ∞ , we have: In particular, the restriction of the distribution of τ on the set T 0 is given by: The GW tree is called critical (resp. sub-critical, super-critical) if µ(p) = 1 (resp. µ(p) < 1, µ(p) > 1). In the critical and sub-critical case, we have that a.s. τ belongs to T 0 . Let P k be the distribution of the forest τ (k) = (τ 1 , . . . , τ k ) of i.i.d. GW trees with offspring distribution p. We set: When there is no confusion, we shall write τ for τ (k) .

Condensation tree.
We say that the offspring distribution p is non-generic if g has convergence radius 1 and µ(p) = g ′ (1) < 1. The corresponding GW tree is also called non-generic.
Assume that p satisfies (1) with µ(p) < 1. Recall the definition of the tree τ * (p) in the introduction. Remark that, as µ(p) < 1, the tree τ * (p) belongs a.s. to T 2 if p is non-generic.
For t ∈ T 0 , x ∈ t, we set: For z ∈ R, we set z + = max(z, 0). Let X be a random variable with distribution p. The following lemma is elementary.
In particular, we have that if x ∈ L 0 (t): Let τ S (p) denote the limit (in distribution) of a critical or sub-critical GW tree τ conditionally on {H(τ ) = n} or {H(τ ) ≥ n} as n goes to infinity. The distribution of τ S (p) is characterized by the properties i) to vii) withp in iv) replaced by the size-biased distribution p • : Remark that, when p is critical, the definitions of τ * (p) and τ S (p) coincide. We have that a.s. τ S (p) belongs to T 1 . Following [1], we notice that the distribution of τ S (p) is characterized by: a.s. τ S (p) ∈ T 1 and for all t ∈ T 0 , x ∈ L 0 (t), Conditioning on the total population size (A = N) We prove Theorem 1.2 for A = N and p non-generic for N. The results of this section appear already in [6] see also [7]. It is a special case of Theorem 1.2 with A = N. We provide here an elementary proof relying on the strong ratio limit property of random walks on the integers. 4.1. The case ρ(p) = 1. We first consider the case ρ(p) = 1 and µ(p) < 1.
Theorem 4.1. Assume that p satisfies (1) and is non-generic for N. We have that: where the limit is understood along the infinite subsequence {n ∈ N * ; P(|τ | = n) > 0}, and: Proof. For simplicity, we shall assume that p is aperiodic, that is P(|τ | = n) > 0 for all n large enough. The adaptation to the periodic case is left to the reader.
and m = |t|. We have: Let (X n , n ∈ N * ) be a sequence of independent random variables taking values in N with distribution p and set S n = n k=1 X k . Let us recall Dwass formula (see [4]): for every k ∈ N * and every n ≥ k, we have Let τ n be distributed as τ conditionally on {|τ | = n}. Using Dwass formula (17), we have We get: . Then use the strong ratio limit property (44) as well as its consequences (45) and (46), to get that: Thanks to (12), we get: Then use Lemma 2.2 to get (15). Since dist (τ |τ | ≥ n) is a mixture of dist (τ |τ | = k) for k ≥ n, we deduce that (16) holds.
Remark 4.2. The proof of (20) also holds if µ = 1. In this case we get in particular that for all t ∈ T 0 and x ∈ L 0 (t): Then the application T(t, x) → P(τ = t)/p(0) can be extended into a probability distribution on T 1 which is given by the distribution of τ * (p) (also equal to the distribution of τ S defined in Remark 3.2). Then use Remark 2.3 to get that dist (τ | |τ | = n) converges to dist (τ * (p)).

4.2.
The case ρ(p) > 1. We consider the case ρ(p) > 1. The offspring distribution p N,θ of (3) has generating function: Recall I N is the set of positive θ for which p N,θ is a well defined probability distribution. Furthermore, according to [9] (see also Proposition 5.5 in [1] for a more general setting), if τ N,θ denotes a GW tree with offspring distribution p N,θ , then the distribution of τ N,θ conditionally on |τ N,θ | does not depend on θ ∈ I N . It is easy to check that µ(p N,θ ) is increasing in θ.
Following [6], we shall say that p is non-generic for N if lim θ↑ρ(p) µ(p N,θ ) < 1. In that case, we have I N = (0, ρ(p)] and p * N defined by (5) is p * N = p N,ρ(p) . Corollary 4.3. Assume that p satisfies (1) and is non-generic for N. We have that: where the limit is understood along the infinite subsequence {n ∈ N * ; P(|τ | = n) > 0}, and: Proof. The first convergence is a direct consequence of (15) and the fact that τ conditionally on {|τ | = n} is distributed as τ N,ρ(p) conditionally on {|τ N,ρ(p) | = n}. The proof of the second convergence is similar to the proof of (16).
This result with Proposition 4.6 and Corollary 5.9 in [1] ends the proof of Theorem 1.2 for the case A = N and gives a complete description of the asymptotic distribution of critical and sub-critical GW trees conditioned to have a large total population size.

Generic and non-generic distributions
Let p be a distribution on N satisfying (1) and let X be a random variable with distribution p. Recall ρ(p) denotes the convergence radius of the generating function g of p. Let A ⊂ N such that p(A) > 0. We consider the modified distribution p A,θ on N given by (3) and let I A be the set of positive θ for which p A,θ is a probability distribution. We have θ ∈ I A if and only if θ > 0 and: In particular, I A is an interval of (0, +∞) which contains 1. We have inf I A = 0 if 0 ∈ A and 1 > inf I A ≥ p(0) if 0 ∈ A. Let: . We deduce from the definition of p A,θ the following rule of composition, for θ ∈ A and θq ∈ A: The generating function, g A,θ , of p A,θ is given by: And we have: Let: , with the convention that inf ∅ = +∞. Notice that the function θ → µ(p A,θ ) is continuous over I A .
Lemma 5.1. Let p be a distribution on N satisfying (1) and A ⊂ N such that p(A) > 0. The function θ → µ(p A,θ ) is increasing over (0, θ c A + ε) I A for some strictly positive ε depending on p. If 0 ∈ A, then the function θ → µ(p A,θ ) is increasing over I A .
Proof. Notice it is enough to consider θ < θ * A . Since p satisfies (1), it is easy to check that p A,θ satisfies (1) for all θ ∈ I A such that θ < θ * A . Thanks to the composition rule, it is enough to prove that θ → µ A,θ is increasing at θ = 1 if µ(p) ≤ 1 + ε for some ε > 0, with p satisfying (1) and ρ(p) > 1.
Let θ ∈ I A . We have: Of course we have h A (1) = 0. The function h A is of class C ∞ on [0, ρ(p)). We obtain: . In particular, we deduce from this last expression that h ′ (1) If 0 ∈ A, then we have inf I A µ(p A,θ ) = p(1)1 A c (1) < 1. If 0 ∈ A, then set q = min I A ∈ (0, 1). Notice that c A (q) = 0 and E q X 1 {X∈A c } = q. Use that the function θ → E θ X 1 {X∈A c } is convex and less than the identity map on (q, 1] to deduce that E Xq X−1 1 {X∈A c } is strictly less than 1. Then use (24) to deduce that: In conclusion, we deduce that inf I A µ(p A,θ ) < 1. Hence, if µ(p) ≥ 1 then Equation (26) has at least one solution.
Lemma 5.4. Let p be a distribution on N satisfying (1) such that µ(p) < 1.
-If 1 < ρ < +∞ and g ′ (ρ) < 1 (and thus g(ρ) < ρ), then p is non-generic for {0} and p is generic for {k} for all k large enough and such that p(k) > 0. Furthermore p is non-generic for A ⊂ N (with p(A) > 0) if and only if: for every non-negative measurable function f . We also have θ * A = ρ. Remark 5.5. We give some consequences and remarks related to the previous Lemma.
(1) If p is generic for {0} then it is generic for all A ⊂ N with p(A) > 0.
(2) If A and B are disjoint subsets of N such that p(A) > 0 and p(B) > 0, then if p is non-generic for A and for B then it is non-generic for A B.  (Let p satisfying (1) be such that ρ(p) > 1 and p non-generic for B = N. Then, according to Lemma 5.4, there exists k large enough such that p(k) > 0 and p is generic for A = {k}.) (5) According to the second part of the proof of Lemma 5.1, we get that there exists n 0 ∈ N * such that: In particular, if p is non-generic for A n 0 then it is non-generic for all A containing 0. (6) Let G be a generating function with radius of convergence ρ G = 1. Let c ∈ (0, 1). Let p be the distribution with generating function: The radius of convergence of g is thus ρ = 1/c and we have: Therefore, we have: If G ′ (1) = 1, then we have G(c) > c. This implies g ′ {0},ρ (1) < g ′ N,ρ (1) = 1. Thus p is generic for N but non generic for {0}.

6.
Vertices with a given number of children I: case 0 ∈ A Assume 0 ∈ A ⊂ N and A = N. Assume that p satisfies (1), µ(p) < 1. We prove Theorem 1.2 for p non-generic for A.
In what follows, we denote by X a random variable distributed according to p. We consider only P(X ∈ A) < 1, as the case P(X ∈ A) = 1 corresponds to A = N of Section 4. For t ∈ T 0 , we set L A (t) = {u ∈ t, k u (t) ∈ A} the set of nodes whose number of children belongs to A and define L A (t) = Card (L A (t)).
For a tree t ∈ T 0 , following [11,14], we can map the set L A (t) onto a tree t A . We first define a map φ from L A (t) on U and a sequence (t k ) 1≤k≤n of trees (where n = L A (t)) as follows. Recall that we denote by < the lexicographic order on U . Let u 1 < · · · < u n be the ordered elements of L A (t).
• For 1 < k ≤ n, set w k = M RCA({u k−1 , u k }) the most recent common ancestor of u k−1 and u k and recall that S w k (t) denotes the tree above w k . We set s = {w k u, u ∈ S w k the subtree above w k and v = min(L A (s)). Then, we set the concatenation of the node φ(v) with the integer k φ(v) (t k−1 ) + 1, and In other words, φ(u k ) is a child of φ(v) in t k and we add it "on the right" of the other children (if any) of φ(v) in the previous tree t k−1 to get t k . It is clear by construction that t k is a tree for every k ≤ n. We set t A = t n . Then φ is a one-to-one map from L A (t) onto t A . The construction of the tree t A is illustrated on Figure 1. Notice that L A (t) is just the total progeny of t A . If τ is a GW tree with offspring distribution p, the tree τ A associated with L A (τ ) is then, according to [14] Theorem 6 (for the particular case 0 ∈ A), a GW tree whose offspring distribution p A is defined as follows. Let N , Y ′′ and (Y ′ k , k ∈ N) be independent random variables such that N is geometric with parameter p(A), Y ′′ is distributed as X conditionally on {X ∈ A} and (Y ′ k , k ∈ N) are independent random variables distributed as X − 1 conditionally on {X ∈ A}. We set: with the convention that ∅ = 0. Then p A is the distribution of X A . Let g A denote its generating function: An elementary computation gives: We recover that if τ is critical (µ(p) = 1) then τ A is critical as µ(p A ) = 1, see also [14] Lemma 6. Notice in particular that for all k ∈ A: and for k ∈ A c : Proof. Since (32) implies p A (0) ≥ p(0) and that µ(p) < 1 with (31) implies µ(p A ) < 1, we deduce that p A satisfies (1).
Let ρ A be the convergence radius of the serie given by E z X 1 {X∈A} and ρ A c be the convergence radius of the series given by E z X 1 {X∈A c } . We get that min(ρ A , ρ A c ) = ρ(p). We deduce that the convergence radius of g A is ρ(p) if ρ(p) = 1 or if ρ(p) > 1 and g ′ (ρ(p)) < 1.
6.1. The case ρ(p) = 1. We state now the main result of this section. Theorem 6.2. Assume that p satisfies (1), µ(p) < 1 and ρ(p) = 1. We have that: where the limit is understood along the infinite subsequence {n ∈ N * ; P(L A (τ ) = n) > 0}, as well as Proof. For simplicity, we shall assume that p A is aperiodic. The adaptation to the periodic case is left to the reader. We define for j ∈ N and n ≥ 2: We have: Let (X n , n ∈ N * ) be independent random variables taking values in N with distribution p A and set S n = n k=1 X k . According to Dwass formula (17), we have: Let τ n be distributed as τ conditionally on {L A (τ ) = n}. Then we have, using (47) and (48): + 1, k), ℓ) . Then use the generalizations of the strong ratio limit properties (44), (50) and (51) to get that: Thanks to (12), we get: lim n→+∞ P(τ n ∈ T + (t, x, k)) = P(τ * (p) ∈ T + (t, x, k)).
6.2. The case ρ(p) > 1. We consider the case p non-generic for A with ρ(p) > 1. In particular, we have g ′ (ρ) < 1 and g(ρ) < ρ thanks to Lemma 5.4. Recall the offspring distribution p A,θ defined by (3). Notice that the normalizing constant c A (θ) is given by: Notice that p A,1 = p. Since ρ(p) is also the convergence radius of g A , see Lemma 6.1, we deduce that p A,θ is well defined for θ ∈ [0, ρ(p)] and θ * A = ρ(p). Let g A,θ be the generating function of p A,θ .
According to [9] if A = {0} and Proposition 5.5 in [1] for the general setting, if τ A,θ denotes a GW tree with offspring distribution p A,θ , then the distribution of τ A,θ conditionally on L A (τ A,θ ) does not depend on θ ∈ [0, ρ(p)]. Remark 6.3. It is easy to check that: . The distribution of τ A,θ is the distribution of τ "shifted" by θ such that the conditional distribution given the number of vertices having a number of children in A is the same. Then, according to (38), the tree (τ A,θ ) A of vertices having a number of children in A associated with τ A,θ is distributed as the distribution of τ A "shifted" by θ such that the conditional distribution given the total number of vertices is the same.
The proof of the following corollary is similar to the one of Corollary 4.3.
Corollary 6.4. Assume that p satisfies (1) and is non-generic for A. Let p * A = p A,ρ(p) . We have that: where the limit is understood along the infinite subsequence {n ∈ N * ; P(L A (τ ) = n) > 0}, as well as This result with Proposition 4.6 and Corollary 5.7 in [1] ends the proof of Theorem 1.2 for the case 0 ∈ A, and gives a complete description of the asymptotic distribution of critical and sub-critical GW trees conditioned to have a large number vertices with given number of children.

7.
Vertices with a given number of children II: case 0 ∈ A Let A ⊂ N. We assume in this section that 0 ∈ A and p(A) > 0. We prove Theorem 1.2 for p non-generic for A. Notice we follow the spirit of the case 0 ∈ A. 7.1. Setting and notations. Although the construction of the previous section also holds in that case with a different offspring distribution, we failed to get analogues to formulas (32) and (33). Therefore, we prefer to map L A (τ ) onto a forest F A (τ ) of independent GW trees. Let us describe this map.
Let t ∈ T 0 . We define a mapφ from L A (t) into the set n≥1 T n 0 of forests of finite trees as follows.
First, for u ∈ t we define S A u (t) the subtree rooted at u with no progeny in A by as the leaves of S A u (t) that belong to A. We setS and we setC A ∅ (t) the set of leaves ofS A ∅ (t) that belong to L A (t). LetÑ ∅ (t) = Card (C A ∅ (t)). Then the range ofφ belongs to TÑ ∅ (t) 0 . Moreover if u 1 < u 2 < · · · < uÑ ∅ (t) are the elements ofC A ∅ (t) ranked in lexicographic order, we set for every where ∅ (i) denotes the root of the i-th tree in TÑ ∅ (t) 0 . We then constructφ recursively: if u ∈ L A (t) andφ(u) = v (i) (which is an element of the i-th tree), then we denote by u 1 < · · · < u k the elements of C A u (t) ranked in lexicographic order and we set for 1 ≤ j ≤ kφ Finally, we set F A (t) =φ(t). Let τ be a Galton-Watson tree with offspring distribution p. Let us describe the distribution of F A (τ ).
We define the offspring distributionp by ThenS A ∅ (τ ) is distributed as a (subcritical) GW tree with offspring distributionp. In particular, if we denote by L the number of leaves ofS A ∅ (τ ), then we have where X is a random variable distributed according to p. Moreover, conditionally given L, the random variable N := N ∅ (τ ) has a binomial distribution with parameter (L, p(A)/(p(0)+ p(A))).
Let X A be the random variable where Z ′ is distributed as X conditionally given {X ∈ A} and (N k , k ∈ N) is a sequence of independent random variables, independent of Z ′ , and distributed as N . We denote by p A the law of X A . Then the forest F A (τ ) is distributed as N independent GW trees with offspring distribution p A .

Main result.
We recall that L A (τ ) is aperiodic since 0 ∈ A, see [1].
Theorem 7.1. Assume that p satisfies (1) and µ(p) < 1 and ρ(p) = 1. We have that: as well as Proof. It is enough to prove that for all t ∈ T 0 , x ∈ t and k ∈ N: x, k)) .
Set M 0 = 0 and M n = n k=1 N k for n ∈ N * . Let m = L A (t) − 1 A (k x (t)) and ℓ = k x (t). Recall (36). We have where we used Dwass formula (17) for the last equality where S n = n k=1 X k with (X k , k ∈ N * ) independent random variables distributed as X A , see also (58). Recall (59). In particular, we have: with: a n,j = n n j Notice that Lemma 8.6 implies that lim n→+∞ a n,j = 1. Then use Lemma 8.9 to get: This ends the proof.
Corollary 7.2. Assume that p satisfies (1), is non-generic for A. Let p * A = p A,ρ(p) . We have that: This result with Proposition 4.6 and Corollary 5.7 in [1] for the generic case ends the proof of Theorem 1.2 for 0 ∈ A and gives a complete description of the asymptotic distribution of critical and sub-critical GW trees conditioned to have a large population. 8. Appendix 8.1. Strong ratio limit property. Let (X n , n ∈ N) be independent random variables taking values in N with distribution p = (p(k), k ∈ N). We assume that: (43) µ(p) ≤ 1 and either µ(p) = 1 or, for all θ > 0, E e θX 1 = +∞.
Let S n = n k=1 X k . We assume that p is aperiodic (that is P(S n = n) > 0 for all n large enough). According to [8] or [12], we have the following strong ratio limit property for all m, k ∈ Z: We deduce the following corollary. Recall the definition of δ 0 n and δ 1 n of (18) and (19). Corollary 8.1. Assume that p satisfies (43) and is aperiodic. For all k ∈ Z and ℓ ∈ N, we have: Proof. Since P(S n+1 = n + ℓ) = j∈N p(j) P(S n = n + ℓ − j), we have: Then use (44) to get (45). Notice that, by exchangeability: j∈N jp(j) P(S n = n + ℓ − j) = E X 1 1 {S n+1 =n+ℓ} = n + ℓ n + 1 P(S n+1 = n + ℓ).
Thus we have: Then use (44) to get: lim Since 1 − j<ℓ jp(j) = 1 − µ(p) + j≥ℓ jp(j), this gives (46). 8.2. Generalization of the strong ratio limit property I. Assume that p satisfies (43) and is aperiodic. Let X be a random variable taking values in N with distribution p. Recall g denote the generating function of p.
Let A ⊂ N such that 0 ∈ A. Let p A be the distribution on N with generating function g A given by (30) and X A distributed according to p A . Recall µ(p A ) is given by (31). In particular µ(p) = 1 (resp. µ(p) ≤ 1) implies µ(p A ) = 1 (resp. µ(p A ) ≤ 1). And from the proof of Lemma 6.1, we get that E e θX = +∞ for all θ > 0 implies that E e θX A = +∞ for all θ > 0.
Let (X n , n ∈ N) be independent random variables, independent of X, taking values in N with distribution p A . Let S n = n k=1 X k . We assume that p A is aperiodic (that is P(S n = n) > 0 for all n large enough). In particular the strong ratio limit property (44) holds as well as (45) and (46) hold with p replaced by p A .
Recall (36), that is n j = n − 1 A (j), and let: and (48) We stress that in (18) and (19), (S n , n ∈ N) is a random walk with increments distributed according to p; whereas in (47) and (48), (S n , n ∈ N) is a random walk with increments distributed according to p A .
Lemma 8.2. Assume that p satisfies (43) and is aperiodic. For all k ∈ Z and ℓ ∈ N, we have: Proof. We define: a n (j) = p(j) P(S n j = n j + ℓ − j) P(S n = n) n n j as well as b n (j) = p A (j) P(S n−1 = n + ℓ − j − 1) P(S n = n) with the convention that p A (−1) = 0. Thanks to the strong ratio limit property (that is (44) with p A instead of p), we have lim n→+∞ a n (j) = p(j) and lim n→+∞ b n (j) = p A (j) + p A (j − 1)/p(A). We have: We deduce from the strong ratio limit property (that is (44) with p A instead of p) that: Then use (32) and (33) to get that a n (j) ≤ 2b n (j) for n ≥ 2 and the dominated convergence theorem to get that: lim n→+∞ j∈N a n (j) = j∈N lim n→+∞ a n (j) = 1.
Notice that j∈N a n (j) = E n n X 1 {X+Sn X =n X +ℓ} /P(S n = n) to deduce that (49) holds. Since δ 0,A n (k, ℓ) = j≥k a n (j), the proof of (50) is then similar to the proof of (45).
For all ℓ ∈ Z, n ≥ 2, we have: Proof. We first prove (52) for ℓ ≤ 0. Let k ≥ 1. By decomposing according to the number of children of the root of the first tree in the forest, we have: with the convention that P 0 (·) = 0. Then using Dwass formula (17) in each side of this equality, we get: Take ℓ = 1 − k to get that (52) holds for ℓ ≤ 0. Unfortunately, we didn't get a similar proof for ℓ ≥ 1 and we prove (52) for ℓ ≥ 1 by induction. Let ℓ ≥ 0. Assume that (52) holds for all ℓ ′ ≤ ℓ and all n ≥ 2, and let us prove it holds for ℓ + 1 and all n ≥ 2. We have: Using (52), we have: So we have: Proof. We shall mimic the proof of the strong ratio limit property provided in [12]. Since p A is aperiodic, the denominator of (56) is positive for n large enough and it is enough to prove the result for m = 1 and k such that The proof will be complete as soon as we prove that: converges to 0 for all ε > 0. Notice that: According to [12], since p A is non-generic with ρ(p A ) = 1, we have lim n→+∞ P(|p A n (k) − p A (k)| > ε)/P(S n = n) = 0. By Fatou and using the strong ratio limit property, we have: Since ε > 0 is arbitrary, we deduce that lim n→+∞ J n = 0.
Remark 8.5. Notice that, from the proof of the lemma, we see that N could be replaced by any non-negative integrable random variable independent of (X k , k ∈ N * ).
Recall that M 0 = 0 and for n ∈ N * : We assume that (N k , k ∈ N * ) and (X k , k ∈ N * ) are independent. We have the following result.
Let p N be the distribution of N . We have, using the decomposition of the GW tree with respect to the descendants of ∅ in A and Dwass formula (17): More generally, we have with N independent of S n and M j−1 .
We set for ℓ ∈ Z: The next lemma is the analogue of Lemma 8.3 in our current setting.
Proof. Recall that E N 1 {Sn+N =n} = P(L A (τ ) = n). Let k ≥ 0. By decomposing τ under P k+1 with respect to the number of children of the first tree in the forest, we get: Then use (58) and Lemma 8.6 to deduce that: lim n→+∞ n P k+1 (L A (τ ) = n) E N 1 {Sn+N =n} = k + 1.
This gives the lemma.
Proof. Notice that nN (X − ℓ) + /n X is integrable. Mimicking the proof of Lemma 8.4 and using that n X takes only two possible values a.s., we get for m, k ∈ Z: lim n→+∞ E n n X N (X − ℓ) + 1 {S n X −m +M X−1−ℓ +N =n X −k} E n n X N (X − ℓ) + 1 {Sn X +M X−1−ℓ +N =n X } = 1.
Then take m = ℓ − k ≥ 0 to get the result.