Local limits of conditioned Galton-Watson trees I: the infinite spine case

We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.


Introduction
Galton-Watson (GW) processes constitute a very simple model of population growth where all individuals give birth independently of each others to a random number of children with the same offspring distribution p. This population growth can be described by a genealogical tree τ that we call the GW tree. It is well-known that in the sub-critical case (the mean number of children of a single individual is strictly less than 1) and in the (non-degenerate) critical case (the mean number of children of an individual is 1) the population becomes a.s. extinct. However, one can define in these two cases a tree τ * with an infinite spine, that we call Kesten's tree in this paper, which can be seen as the tree conditioned on non-extinction, defined as the local limit in distribution of the tree τ conditioned to reach height n, when n tends to infinity, see Kesten [16]. This result is recalled here in Section 2.4. The tree τ * happens to be the size-biased tree already studied earlier, see e.g. Hawkes [10], Joffe and Waugh [13] as well as Lyons, Pemantle and Peres [19]. It also appears (for GW processes only) as a Q-process and can be viewed as a GW tree with immigration, see Athreya and Ney [4]. We want to stress that we only consider here local limits i.e. we look at the trees up to a fixed height h. Other limits can be considered such as scaling limits of conditioned GW trees (see [7,18,24]) but this is not the purpose here.
It is also known that, at least in the critical case, other conditionings such as conditioning by the total progeny, see Kennedy [15] and Geiger and Kaufmann [9], or by the number of leaves, see Curien and Kortchemski [6], lead to the same local limit in distribution. See also the survey from Janson [12].
For all those cases, the conditioning event can be written as {τ ∈ A n } with A n of the form: where A : t → A(t) is a functional defined on the set of trees and satisfying an additive property, see Equation (9). The main result of this paper, see Theorem 3.1 for a precise statement, unifies all the previous conditionings and gives a necessary and sufficient condition to obtain Kesten's tree as a limit. In the non-degenerate critical case, if A satisfies the additive property (9) , then the following two statements are equivalent (with some additional aperiodic condition for the converse): • lim n→+∞ P(τ ∈ A n+1 )/P(τ ∈ A n ) = 1, • The distribution of τ conditionally on {τ ∈ A n } converges to the distribution of Kesten's tree τ * . Using this result, we give elementary proofs for the convergence in distribution to Kesten Let us mention that assertion (i) with the conditioning event {H(τ ) ≥ n} was first proved by Kesten [16] in the critical case under a finite variance condition, and in [12], Theorem 7.1, in full generality. Property (ii) is also proved in full generality in [12], Theorem 7.1 (the sub-critical case is also studied in [12], see the discussion below). Finally, assertion (iii) with the conditioning event {L 0 (τ ) = n} has been proved by Curien and Kortchemski [6], Theorem 4.1, in the critical and finite variance case only.
In fact the conditioning on the large total population size or on the large number of leaves are particular cases of conditioning trees on large number of individuals with a given number of children. This corresponds to the functional A(t) = L A (t) which gives the total number of individuals of the tree t whose number of children belongs to a given set A of nonnegative integers. Such conditioning has already been studied by Rizzolo [24], see also Mimami [20], but for global scaling limits and not local limits. We obtain the convergence in distribution to Kesten's tree τ * of a critical GW tree without any additional moment condition on the offspring distribution, conditioned on: (iv) Large number of individuals with number of children in a given set A (critical case), with A(t) = L A (t) and conditioning event {L A (τ ) = n} or {L A (τ ) ≥ n}. Here, we use the fact that L A (τ ) is distributed according to the total progeny of another critical GW tree, which allows to use (ii), see [20,24]. Let us remark that the total progeny (A = N), the number of leaves (A = {0}) and the number of internal nodes (A = N \ {0}) are particular cases of this conditioning.
The main ingredients in the proof for (ii), (iii) and (iv) are Dwass formula for the representation of the total progeny of a GW tree using random walks, and the strong ration theorem for these random walks which has some links with the local sub-exponential property of the total progeny of GW trees, see [3].
We then study the subcritical case and define a one-parameter family (p θ , θ ∈ I) of distributions on the set of integers such that the GW tree τ associated with the offspring distribution p and the GW tree τ θ associated with the offspring distribution p θ have the same conditional distributions given L A , see Proposition 5.5. This generalizes Kennedy's transformation [15] concerning the total progeny, and the pruning of Abraham, Delmas and He [2] concerning the number of leaves. According to [12], we say that p is generic (with respect to A) if there exists θ c such that p θc is critical. We then immediately deduce, see Corollary 5.7, that if p is generic, then the distribution of τ conditionally on {L A (τ ) = n} (in the aperiodic case) or on {L A (τ ) ≥ n} converges to the distribution of the Kesten's tree τ * θc associated with the critical offspring distribution p θc . When there is no such θ c , then a condensation phenomenon may appear: Jonsson and Stefansson [14] or [12] proved for the conditioning on the total progeny that the limiting tree in that case is not Kesten's tree but a tree with a unique node with an infinite number of offsprings. We shall investigate this condensation phenomenon for other conditionings in a forthcoming paper [1]. Let us add that an example is given in [1] of an offspring distribution which is generic with respect to a set A and non-generic with respect to another set. Hence, it seems difficult to give a sufficient condition for an offspring distribution to be generic (i.e. to have existence of the critical value θ c ).
Finally, we consider another conditioning which does not enter in the framework of Theorem 3.1 : conditioning on the size on the n-th generation. However, we can adapt the proof of Theorem 3.1 to get an analogous result in that case, see Proposition 6.1. We apply this result to a critical geometric offspring distribution where explicit computations can be performed to prove that the corresponding GW tree conditioned on the n-th generation being positive but smaller that n 2 converges in distribution to Kesten's tree. Using results on local limit of GW processes from Nagaev and Vakhtel [21,22], this result can be extended to very general critical offspring distributions.
The paper is organized as follows. In Section 2, we recall the framework we use for discrete trees and define the GW tree τ and Kesten's tree τ * associated with offspring distribution p. In Section 3, we state and prove the necessary and sufficient condition for convergence in distribution of the conditioned tree to Kesten's tree. We apply this result in Section 4 to recover the classical results on critical conditioned GW trees and we study in Section 5 the case of the number of individuals with out-degree in a given set for the critical and sub-critical case. Finally, we study in Section 6 the conditioning on the size of the n-th generation of the GW tree. If K is a subset of N * , we call the span of K the greatest common divisor of K. If X is an integer-valued random variable, we call the span of X the span of {n > 0, P(X = n) > 0} the restriction to N * of its support.
The most recent common ancestor of a subset s of U , denoted by M(s), is the unique element u of u∈s A u with maximal length |u|.
For u, v ∈ U , we denote by u < v the lexicographic order on U i.e. u < v if u ∈ A v or, if we set w = M(u, v), then u = wiu ′ and v = wjv ′ for some i, j ∈ N * with i < j.
A tree t is a subset of U that satisfies: • If u ∈ t, then A u ⊂ t.
• For every u ∈ t, there exists a non-negative integer k u (t) such that, for every positive integer i, ui ∈ t iff 1 ≤ i ≤ k u (t). The integer k u (t) represents the number of offspring of the vertex u ∈ t. The vertex u ∈ t is called a leaf if k u (t) = 0. The vertex ∅ is called the root of t. Let us remark that, for a tree t, we have Let t be a tree. The set of its leaves is L 0 (t) = {u ∈ t; k u (t) = 0}, its height is defined by and can be infinite. For u ∈ t, we define the sub-tree S u (t) of t "above" u as: We denote by T the set of trees, by ≤ h} the subset of trees with height at most h ∈ N, and by the subset of trees with a unique infinite spine. Notice that T 0 and T (h) are countable and T 1 is uncountable as the set of infinite sequences of positive integers can be embedded in T 1 . For h ∈ N the restriction function r h from T to T is defined by: We endow the set T with the ultrametric 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.
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.
and h ∈ N and notice that: Since the distance is ultrametric, 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 [5] 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 dist for the convergence in distribution of the sequence (T n , n ∈ N * ) to T . We deduce from the portmanteau theorem that the sequence (T n , n ∈ N * ) converge in distribution to T if and only if for all h ∈ N, t ∈ T (h) : For t ∈ T and u ∈ t, set k u (t) = −1. The convergence in distribution of the sequence (T n , n ∈ N * ) to T is also equivalent to the finite dimensional convergences in distribution of the sequence ((k u 1 (T n ), . . . , k um (T n )), n ∈ N * ) to (k u 1 (T ), . . . , k um (T )) for all m ∈ N * and u 1 , . . . , u m ∈ U .
As we shall only consider T 0 -valued random variables that converge in distribution to a T 1 -valued random variable, we shall give an alternative characterization of convergence in distribution that holds for this restriction. To present this result, we introduce some notations. If t, s ∈ T and x ∈ L 0 (t) we denote by: the tree obtained by grafting the tree s on the leaf x of the tree t. For every t ∈ T and every x ∈ L 0 (t), we shall consider the set of trees obtained by grafting a tree on the leaf x of t: It is easy to see that T(t, x) is closed. It is also open, as for all s ∈ T(t, x) we have that Moreover, notice that the set T 1 is a Borel subset of the set T.
Lemma 2.1. 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 for every t ∈ T 0 and every x ∈ L 0 (t), we have: in the other cases.
For every h ∈ N and every t ∈ T (h) , we have that t ′ belongs to r −1 h ({t}) T 1 if and only if t ′ belongs to some T(s, x) where x is a leaf of t such that |x| = h and s belongs to r −1 h ({t}) T 0 such that x is also a leaf of s. Since T 0 is countable, we deduce that F generates the Borel σ-field on T 0 ∪ T 1 . In particular F is a separating class on T 0 T 1 .
Since A ∈ F is closed and open as well, according to Theorem 2.3 of [5], to prove that the family F is a convergence determining class, it is enough to check that for all t ∈ T 0 T 1 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 (4) is satisfied. This proves that the family F is a convergence determining class on T 0 T 1 .
Since, for t ∈ T 0 and x ∈ L 0 (t) the sets T(t, x) and {t} are open and closed, we deduce from the portmanteau Theorem that if (T n , n ∈ N * ) converges in distribution to T , then (3) holds for every t ∈ T 0 and every x ∈ L 0 (t).
2.3. GW trees. Let p = (p(n), n ∈ N) be a probability distribution on the set of the nonnegative integers. We assume that 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 every 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 = 1 (resp. µ < 1, µ > 1).

2.4.
Conditioning on non-extinction. Let p be an offspring distribution satisfying Assumption (5) with µ ≤ 1 (i.e. the associated GW process is critical or sub-critical). We denote by p * = (p * (n) = np(n)/µ, n ∈ N) the corresponding size-biased distribution. We define an infinite random tree τ * (the size-biased tree that we call Kesten's tree in this paper), whose distribution is as follows. There exists a unique infinite sequence (V k , k ∈ N * ) of positive integers such that, for every h ∈ N, V 1 · · · V h ∈ τ * , with the convention that The joint distribution of (V k , k ∈ N * ) and τ * is determined recursively as follows: for each h ∈ N, conditionally given (V 1 , . . . , V h ) and r h (τ * ), we have: • The number of children (k v (τ * ), v ∈ τ * , |v| = h) are independent and distributed according to Notice that by construction, τ * ∈ T 1 a.s. Following Kesten [16], the random tree τ * can be viewed as the tree τ conditioned on non-extinction as: As a direct consequence we get that for all h ∈ N, t ∈ T (h) , u ∈ t such that |u| = h: and for all t ∈ T 0 , x ∈ L 0 (t): (7) P(τ * ∈ T(t, x)) = µ −|x| P(τ ∈ T(t, x)).

Main result
Let A be an integer-valued function defined on T which is finite on T 0 and satisfies the following additivity property: there exists an integer-valued function D defined on T such that, for every t ∈ T 0 , every x ∈ L 0 (t) and for everyt such that A(t ⊛ (t, x)) is large enough, Let n 0 ∈ N ∪ {+∞} be given. We define for all n ∈ N * , the subset of trees Common values of n 0 that will be considered are 1 and +∞.
The following theorem states that the distribution of the GW tree τ conditioned to be in A ∞ , the limit of A n , is distributed as τ * as soon as the probability of A n satisfies some regularity. We denote by dist (τ |τ ∈ A n ) the conditional law of τ given {τ ∈ A n }.
Theorem 3.1. Assume that Assumptions (5) and (9) hold, that P(τ ∈ A n ) > 0 for n large enough and that one of the two following conditions • µ = 1 or • µ < 1 and D(t, x) = |x| for all t ∈ T 0 , x ∈ L 0 (t). Then, if and if the span of {D(t, x); t ∈ T 0 and x ∈ L 0 (t)} N * is one, then (10) holds.
Recall that the local convergence in distribution towards τ * is equivalent to Proof. Let us first remark that, as we supposed that µ ≤ 1, we have a.s. τ ∈ T 0 and thus we are in the setting of Lemma 2.1. Using (6), we have for every t ∈ T 0 , x ∈ L 0 (t) andt ∈ T 0 : Let t ∈ T 0 and x ∈ L 0 (t). Then, if n is large enough so that we can apply Equation (9), we get: where we used (8) for the last equality. Therefore we have Then, using (10) and that D(t, x) = |x| if µ < 1, we obtain that: For all t ∈ T 0 and all n > A(t), we have and thus: We deduce from Lemma 2.1 that (11) holds. Conversely, if (11) holds, then Lemma 2.1 implies that (13) and (14) hold. The fact that the span of {D(t, x); t ∈ T 0 and x ∈ L 0 (t)} N * is one and (12) imply, with Bezout theorem, that (10) holds.

4.1.
Conditioning on extinction after large time. We give here a simple proof of Kesten's result for the convergence in distribution of a critical or sub-critical GW tree conditioned on non-extinction, see [16] under a finite variance condition and [12] for the general case.
Proposition 4.1. Let τ be a critical or sub-critical GW tree with offspring distribution p satisfying Assumption (5). Then, we have Proof. Consider A(t) = H(t) and n 0 = +∞ that is A n = {t ∈ T; H(t) ≥ n}. Notice that in this case for a treet such that H(t) is larger than H(t), we have for every x ∈ L 0 (t) Therefore, Condition (9) is satisfied by A. According to Theorem 3.1, it suffices to prove to get (15).
We denote by ϕ the generating function of p and we define recursively ϕ 1 = ϕ and for n ≥ 1, ϕ n+1 = ϕ n • ϕ. As ϕ n is the generating function of the distribution of {u ∈ τ ; |u| = n} the number of individuals at height n, we have P(τ ∈ A n ) = 1 − ϕ n (0). We also have lim n→+∞ ϕ n (0) = 1 and

4.2.
Conditioning on extinction at large time.

4.3.
Conditioning on the total population size, critical case. We recover here results from Theorem 7.1 in [12] on the convergence in distribution of a critical GW tree conditioned on the size of its total progeny to Kesten's tree. Our proof is based on Dwass formula (see [8]) that we recall now. Let (τ k , k ∈ N * ) be independent GW trees distributed as τ . Set W k = Card (τ k ). Let (X k , k ∈ N * ) be independent integer-valued random variables distributed according to p. For k ∈ N * and n ≥ k, we have: (20) P(W 1 + . . . + W k = n) = k n P(X 1 + . . . + X n = n − k).
We also recall some results on random walks. Let Y be an integrable random variable taking values in Z, such that E[Y ] = 0, P(Y = 0) < 1 and the span of |Y | is 1. We consider the random walk S = (S n , n ∈ N) defined by: (21) S 0 = 0 and S n = n k=1 Y k for n ∈ N * .
Then the random walk S is recurrent. We define the period of S as the span of the set {n > 0, P(S n = 0) > 0}. If S is aperiodic (i.e. has period 1), the strong ratio theorem for recurrent aperiodic random walks, see Theorem T1 p49 of [25], gives that, for ℓ ∈ Z: If S has period d, then for all k ∈ {1, . . . , d}, there exist j k ∈ Z and n k ∈ N * such that (23) ∀n ≥ n k , P(S nd+k = j k ) > 0.
The strong ratio theorem can then easily be adapted to get that, for ℓ ∈ Z, k ∈ {1, . . . , d}: Notice that (20) and (24) directly imply that the total progeny distribution enjoys the local sub-exponential property, see [3]. Then we have and Remark 4.4. If we consider A(t) = Card (t) and n 0 = +∞ that is A n = {t ∈ T, Card (t) ≥ n}, the converse of Theorem 3.1 gives the sub-exponential property: Proof of Proposition 4.3. Consider A(t) = Card (t) and n 0 = d. Then we have We have for every t ∈ T, without any additional assumption, so Condition (9) holds. Again, it therefore suffices to prove (29) lim n→+∞ P(Card (τ ) ∈ [n + 1, n + 1 + d)) P(Card (τ ) ∈ [n, n + d)) = 1 to get (25). By the definition of d, a.s. we have A(τ ) ∈ dN + 1. We consider an integer valued random variable X distributed according to p and we set Y = X − 1 so that E[Y ] = 0 since we supposed that µ = 1. The random walk defined by (21) has period d and we can choose j 1 = −1 in (23) as P(Y = −1) > 0. Dwass formula (20) implies that, for k = ⌊(n − 1)/d⌋: Using (24), we deduce that: which readily implies (29). The second assertion (26) is then a straightforward consequence of (29).
Remark 4.5. Notice that the local limit theorem gives asymptotics for P(S n = −1) when the distribution of X belongs to the domain of attraction of a stable law, see Theorem 4.2.1 of [11] or Theorem 1.10 in [17]. This gives asymptotics for P(τ ∈ A n ) which in turns allow to recover Condition (10).

4.4.
Conditioning on the number of leaves, critical case. For a finite tree t ∈ T 0 , we denote by L 0 (t) = Card (L 0 (t)) the number of leaves of t. The next proposition (which seems to be a new result) is in fact a particular case of the proposition of the next section. However, we prove it separately for methodological purpose as its proof and in particular the construction of the GW tree that codes L 0 (t) of Remark 4.8 are much simpler in that particular case.
Proposition 4.6. Let τ be a critical GW tree with offspring distribution p satisfying Assumption (5). Let d 0 be the span of the random variable L 0 (τ ) − 1. Then we have and Proof. We consider A(t) = L 0 (t) and n 0 = d 0 which yields A n = {t ∈ T; L 0 (t) ∈ [n, n + d 0 )}. We have for every trees t,t ∈ T 0 and every x ∈ L 0 (t) According to [20], see also Remark 4.8 below, L 0 (τ ) is distributed as the total size of a critical GW tree τ 0 with offspring distribution given by the distribution of: with (Z k , k ∈ N * ) and N independent random variables such that (Z k , k ∈ N * ) are independent and distributed as X − 1 conditionally on {X ≥ 1} (where X is a random variable distributed according to p) and N has a geometric distribution with parameter p(0). As E[X 0 ] = 1, we get that τ 0 is critical. Notice that d 0 is also the span of the random variable X 0 . It follows from (29) that: (34) lim n→+∞ P(L 0 (τ ) ∈ [n + 1, n + 1 + d 0 )) P(L 0 (τ ) ∈ [n, n + d 0 )) = 1.
Then use Theorem 3.1 to get that (30) holds. If we consider n 0 = +∞ that is: arguing as in the proof of the second part of Proposition 4.3, we get (31).
Remark 4.8. We shall briefly recall how one can prove that L 0 (τ ) is distributed as the total size of a GW process by mapping the set of leaves L 0 (τ ) onto a GW tree, see [20,24] for details. Let t be a tree. For u ∈ t, we define the left branch starting from u as: We also define the left leaf G(u) of u and the left ancestors A g (v) of a leaf v as: For a leaf v ∈ L 0 (t), we define its leaf-children as: , labeled according to the following order: G t (ui) < G t (u ′ i ′ ) if u < u ′ in the lexicographic order or if u = u ′ and i < i ′ . This defines a tree, obtained from the leaves of t, denoted by If τ is a GW tree then τ {0} = F {0} (τ ) is also a GW tree with offspring distribution given by the distribution of X 0 in (33).

Conditioning on the number of individuals having a given number of children
Let A be a non-empty subset of N. For a tree t ∈ T, we write L A (t) = {u ∈ t; k u (t) ∈ A} the set of individuals whose number of children belongs to A and L A (t) = Card (L A (t)) its cardinal. The case A = {0} represents the set of leaves of t and has been treated in Section 4.4. We can also have L A (t) = Card (t) by taking A = N or L A (t) can also be the number of internal nodes by taking A = N * .
We set: p(A) = k∈A p(k).

5.1.
The critical case. Let us first remark that for every t ∈ T 0 , every x ∈ L 0 (t) and everỹ t ∈ T and hence L A satisfies the additive property (9) with D(t, x) = L A (t) − 1 {0∈A} .
Theorem 5.1. Let τ be a critical GW tree with offspring distribution p satisfying Assumption (5) and such that p(A) > 0. Let d A be the span of the random variable L A (τ ) − 1. Then we have Remark 5.2. It is interesting to note that previous works [24,17] studying conditioned GW trees involving L A required additional assumptions on the moments of p or on A (finite variance offspring distribution and 0 ∈ A in [24], and offspring distribution p in the domain of attraction of a stable law with either A or N \ A finite in the case of infinite variance offspring distribution in [17]).
Remark 5.3. In the proof of Theorem 5.1, we will see that if 0 ∈ A, then d A = 1.
Remark 5.4. As a corollary, we get the following result, which is proven using the same technique as in Remark 4.4: Proof of Theorem 5.1. 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 the critical case with A = N of Section 4.3.
For a tree t such that L A (t) = ∅, following [24], 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, recall that S M ({u k−1 ,u k }) (t) denotes the tree above the most recent common ancestor of u k−1 and u k , and we set 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  2. If τ is a GW tree with offspring distribution p, the tree τ A associated with L A (τ ), conditioned on L A (τ ) = ∅, is then, according to [24] Theorem 6, a GW tree whose offspring distribution is the law of the random variable X A defined as follows: • Let (X i , i ≥ 1) be a sequence of independent random variables distributed according to p.
Then, L A (τ ) is just the total progeny of τ A . Remark that d A is also the span of X A . Remark that, if 0 ∈ A, then L A (τ ) > 0 and thus q = 1 and X A =X. Notice that we may have d A > 1. On the contrary, if 0 ∈ A, we have q < 1 and therefore P(X A = 1) > 0. As a consequence, we have d A = 1.
Consider n 0 = d A which gives As L A (τ ) , conditioned on being positive, is distributed as the total size of a critical GW tree, we deduce from Subsection 4.3 that and thus by Theorem 3.1 that (35) holds.

5.2.
The sub-critical case. Let p be an offspring distribution. Let A ⊂ N such that p(A) > 0. For every θ > 0 such that k∈N θ k p(k) is finite, we define on N the function p θ by where the normalizing constant c A (θ) is given by: We denote by I the set of θ such that p θ defines a probability distribution on N. Notice that I is an interval with bounds θ 0 < 1 ≤ θ 1 . We have the special cases θ 0 = 0 if 0 ∈ A and Proposition 5.5. Let τ be a GW tree with offspring distribution p satisfying p(0) > 0 and p(0) + p(1) < 1. Let A ⊂ N such that p(A) > 0. For every θ ∈ I, let τ θ be a GW tree with offspring distribution p θ . Then the conditional distributions of τ given {L A (τ ) = n} and of τ θ given {L A (τ θ ) = n} are the same.
Remark 5.6. This proposition covers Kennedy's result [15] for A = N and the pruning procedure of [2] for A = {0}.
Proof. Let t ∈ T 0 . Then we have, using the definition of p θ and (2): We deduce that and finally, for every t ∈ T 0 such that L A (t) = n, we have We shall say that the offspring distribution p is generic (with respect to A) if there exists θ c ∈ I such that p θc is critical.
Corollary 5.7. Let τ be a sub-critical GW tree with offspring distribution p satisfying Assumption (5). Let A ⊂ N such that p(A) > 0. For every θ ∈ I, let τ θ be a GW tree with offspring distribution p θ . If p is generic, that is there exists θ c ∈ I such that p θc is critical, then Remark 5.8. The first convergence of the corollary remains valid for a super-critical offspring distribution but not the second one as the conditional distribution cannot be written as a mixture of the first one as the tree may be infinite.
Remark 5.9. If the critical value θ c of Corollary 5.7 does not exist, then we observe a condensation phenomenon: the limiting tree does not have an infinite spine, but exhibits a unique vertex with an infinite number of children, see [12] for A = N and the forthcoming paper [1] for the general case.
Then use Proposition 6.1 to conclude. Remark 6.3. As for Theorem 3.1, we can obtain the converse of Proposition 6.1. We deduce that, in the geometric case of Corollary 6.2, the GW tree τ conditioned on {G n (τ ) = k⌊n a ⌋}, with k ∈ N * , converges in distribution to Kesten's tree if and only if a ∈ [0, 2). Let X be a random variable with distribution p, d the span of X and set B = E[X(X − 1)]. We recall the theorem of [22]. Assume that p is critical, that Assumption (5) holds and that B is finite. If We also recall Theorem 1 of [21]. Let ρ be the convergence radius of the generating function of p. Assume that p is critical, that Assumption (5) holds and that ρ > 1. Assume also that (42) lim n→+∞ α n n = +∞ and lim n→+∞ α n n 2 = 0. Then there exists c ∈ R such that: lim n→+∞ B 2 n 2 e 2dαn Bn +c αn n 2 log(αn/n) P(G n (τ ) = dα n ) = 4d.
Then using Proposition 6.1, we give an immediate extension of Corollary 6.2 to a large class of offspring distributions. Proposition 6.4. Let p be a critical offspring distribution satisfying Assumption (5) and such that B is finite. Assume either that (α n , n ∈ N) is a sequence of positive integers satisfying (41) or that ρ > 1 and (α n , n ∈ N) is a sequence of positive integers satisfying (42). Let τ be a critical GW tree with offspring distribution p. Then we have dist (τ |G n (τ ) = dα n ) −→ n→+∞ dist (τ * ).