Asymptotic properties of expansive Galton-Watson trees

We consider a super-critical Galton-Watson tree whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau$n distributed as $\tau$ conditioned on the n-th generation, Zn, to be of size an $\in$ N. We identify the possible local limits of $\tau$n as n goes to infinity according to the growth rate of an. In the low regime, the local limit $\tau$ 0 is the Kesten tree, in the moderate regime the family of local limits, $\tau$ $\theta$ for $\theta$ $\in$ (0, +$\infty$), is distributed as $\tau$ conditionally on {W = $\theta$}, where W is the (non-trivial) limit of the renormalization of Zn. In the high regime, we prove the local convergence towards $\tau$ $\infty$ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits ($\tau$ $\theta$ , $\theta$ $\in$ [0, $\infty$]).


Introduction
The study of Galton-Watson (GW) processes and more generally GW trees conditioned to be non extinct goes back to Kesten [24], see Lemma 1.14 therein. In the sub-critical and non-degenerate critical case the extinction event E being of probability one, there are many non equivalent limiting procedures to define a GW tree conditioned on the non-extinction event. Those so-called local limits of GW trees have received a renewed interest recently because of the possibility of condensation phenomenon: a node in the N) the corresponding GW process, with Z n being the size of τ at generation n, starting at Z 0 = 1. Let a ∈ N and b ∈N be respectively the lower and upper bound of the support of p. We have a < b as p is non-degenerate. Let c = P(E) be the probability of the extinction event. We recall that c ∈ [0, 1) is the only root of f (r) = r on [0, 1). Notice that c = 0 if and only if a ≥ 1. When P(Z n = a n ) > 0, we denote by τ n a random tree distributed as τ conditioned on {Z n = a n }. We study the local convergence in distribution of (τ n , n ∈ N * ) according to the rate of growth of the sequence (a n , n ∈ N * ). According to Seneta [33] or Asmussen and Hering [6], we shall consider the Seneta-Heyde norming (c n , n ∈ N) which is a sequence such that Z n /c n converges a.s. to a limit W and P(W = 0) = c, see its definition in Section 4. When µ = +∞, then such a normalization does not exists and when the L log(L) condition holds, that is k∈N * p k log(p k ) < +∞, then c n is equivalent to µ n up to an arbitrary positive multiplicative constant, see Seneta [34]. However, we stress that the L log(L) condition is not assumed in this paper and that we only consider the case µ finite. It is well known that the distribution of W , restricted to (0, +∞), has a continuous positive density w with respect to the Lebesgue measure, see the seminal work of Harris [20] and the general result from Dubuc [13]. However, w is explicitly known in only two cases: the geometric offspring distribution, see Section 1.4 below and the example developed by Hambly [19].
We now introduce the possible local limiting trees.
If b = +∞ and R c > 1, we denote by τ ∞ an inhomogeneous Galton-Watson tree with offspring distribution at generation h given bỹ • Low regime: lim n→∞ a n /c n = 0 and a n > 0 for all n ∈ N * . Then, we have: • Moderate regime: lim n→∞ a n /c n = θ ∈ (0, +∞). Then, we have: such that p(b) > 0 (take b = b if b < ∞), there exists d ∈ N such that for all k ∈ N * and x ∈ [[ka n + d, kb n − d]] with x = kr n 0 (mod L 0 ), we have P k (Z n = x) > 0. Taking k = 1 and x = a n , this provides sufficient conditions for τ n to be well defined. In particular, notice that there exist sequences (a n , n ∈ N * ) in all the regime such that τ n is well defined.
Moreover, we have the following continuity result in distribution for the family of limiting trees. Theorem 1.5. Let p be a non-degenerate super-critical offspring distribution with finite mean. The family (τ θ , θ ∈ [0, +∞)) is continuous for the local convergence in distribution.
In particular, we have The continuity of (τ θ , θ ∈ [0, +∞)) is proven in Section 7 and more precisely in Corollary 7.1 for the continuity at 0. The continuity at 0 allows to explain and extend Corollary 3 from Berestycki, Gantert and Mörters [9] on the convergence in distribution of τ (ε) (distributed as τ conditionally on {0 < W ≤ ε}) towards τ 0 as ε goes down to 0, see Corollary 7.2.
When θ goes to infinity, we only have the following partial results. Theorem 1.6. If b < ∞ or if p is geometric, then we have: Assume b = +∞ and R c > 1. If τ θ converges in distribution as θ goes to infinity, then the limit is τ ∞ .
The continuity at infinity is proven in [1] for the geometric case and in Proposition 8.10 for the Harris case. The fact that τ ∞ is the only possible limit if b = +∞ and R c > 1 is proven in Corollary 8. 8. We conjecture that this convergence indeed holds true.
If R c = 1, then we have no hint concerning the existence or non-existence of possible limits for τ θ as θ goes to infinity. Notice that it is not clear that τ θ is stochastically non-decreasing with θ. Remark 1.7. Partial results concerning the sub-critical case are presented in Section 9, under the assumption that R c > 1 and the equation f (r) = r has a finite root in (1, R c ].
This assumption is equivalent to assuming that the sub-critical GW tree is distributed as a super-critical GW tree conditioned on the extinction event. In this case, we can use the previous results in the super-critical case to get results in the sub-critical case.
To finish with the description of all the possible limiting trees, let us mention that all trees τ θ , for θ ∈ [0, +∞], can be described as an infinite backbone of immortal individuals on which are grafted finite trees distributed as τ 0,0 , i.e. GW trees conditioned on extinction. This description also arises when conditioning a super-critical Galton-Watson tree on survival, see [27], Section 5.7. The finite grafted trees in this case are distributed as τ 0,0 . This is however not always the case. For instance in [4], Section 5.2, the local limit of sub-critical GW trees conditioned on their total progeny to be very large, is an infinite spine on which are grafted independent finite GW trees which are not distributed as τ 0,0 . In our present context, it appears that conditioning on {Z n > 0} or {Z n = a n } affects only the immortal backbone and not the distribution of the grafted finite trees.

Strong ratio theorem for super-critical GW process
We set for k, h ∈ N * : H n (h, k) = P k (Z n−h = a n ) P(Z n = a n ) , where Z is under P k a GW process starting from Z 0 = k. The proofs of Theorem 1.2, when there is no condensation, rely on the elementary identity (2.6) which states that P(r h (τ n ) = t) = H n (h, z h (t))P(r h (τ ) = t), where r h (s) denotes the restriction of the tree s up to generation h ∈ N * , and t is a tree with height h (that is z h (t) > 0 and z h+1 (t) = 0). Since the local convergence in distribution of τ n towards a tree with finite nodes is equivalent to the convergence of P(r h (τ n ) = t) for all h ∈ N * and all tree t of height h, up to the identification of the limit, the local convergence can be deduced from the convergence as n goes to infinity of H n (h, k) for all h, k ∈ N * . The result is in the same spirit as the strong ratio theorem for random walks.
In the next theorem, completing known results described in the discussion below, we explicit all the possible limits of H n (with only partial results in the high regime). Notice that all the regimes described in the following theorem are valid thanks to Remark 1.4. Theorem 1.8. Let p be a non-degenerate super-critical offspring distribution with finite mean µ ∈ (1, +∞). We assume that the sequence (a n , n ∈ N * ) is such that P(Z n = a n ) > 0 for all n ∈ N * .
• Extinction case: a n = 0 for all n ≥ n 0 for some n 0 ∈ N * . If c = 0, then P(Z n = 0) = 0 for all n ∈ N, and thus H n is not defined. If c > 0, then we have: (1.2) • Low regime: lim n→∞ a n /c n = 0 and a n > 0 for all n ∈ N * . We have 1 : • Moderate regime: lim n→∞ a n /c n = θ ∈ (0, +∞). We have, with the notation w k (θ) = where (L 0 , r 0 ) is the type of p. • High regime: lim n→∞ a n /c n = +∞. (Partial results.) We have: (1.5) Contrary to the short proof of the strong ratio theorem for random walks given by Neveu [29], the proof presented here for the strong ratio theorem rely on explicit equivalent of P k (Z n−h = a n ) for n large. The well known extinction case is given in The result for the low regime is much more delicate. We shall distinguish between the Schröder case f (c) > 0 and the Böttcher case f (c) = 0, and in those two cases consider the sequence (a n , n ∈ N * ) bounded or unbounded. The case a n bounded and a = 0 can be found in Papangelou [31]. The case a n bounded, a = 1 is an easy extension of [31], see Case I in the proof of Proposition 6.5 in the Schröder case. The case a n unbounded and a ≤ 1 (Schröder case) can be derived, see Lemma 6.6, from the precise asymptotics of P (Z n = a n ) given by Fleischmann and Wachtel [17]. The case a ≥ 2 (Böttcher case) is given in Lemma 12.4 and Lemma 12.5. The former lemma relies on a precise approximation of P (Z n = a n ) given in Lemma 12.3 for a n unbounded, which is an extension of the precise asymptotics given by Fleischmann and Wachtel [18]. The moderate regime is a direct consequence of the local limit theorem in Dubuc and Seneta [12], see Lemma 6.1 here.
The high regime in the Harris case when lim sup n→∞ a n /b n < 1 is detailed in Lemma 11.6 with = 1. It relies on techniques similar to those developed in [18] or in Flajolet and Odlyzko [16] to get an equivalent to P k (Z n = a n ), see Lemma 11.5. The proof is however given in details because the adaptation is not straightforward. The high regime for the geometric offspring distribution is given in [1].
If b = ∞ and f (R c ) = +∞, we conjecture that τ n converges locally in distribution towards a limit τ ∞ whose root has an infinite number of children. Using the elementary identity (2.6), we deduce the following conjecture that if b = ∞ and f (R c ) = +∞, then: (1.6) If b = +∞ and f (R c ) < +∞, then τ ∞ has no condensation and thus H ∞ (h, k) might exists and be given by f −h+1 (R c ) k /f (R c ), where, for n ∈ N * , f n denotes the n-th iterate of f and f −n its inverse (which is well defined because f n is increasing). See the martingale term in the right hand side of (8.7) with λ = λ c .
If R c = 1, the possible existence of a limit for H n is an open question. See Wachtel, Denisov and Korshunov [35] for a first step in the study of this so-called heavy-tailed case.

Link with the Martin boundary of super-critical GW process
Recall that Z is a super-critical GW process with non-degenerate offspring distribution p with finite mean µ. The Martin boundary M of the non-negative space-time GW process corresponds to all extremal non-negative space-time harmonic functions H defined on N 2 , and is related to the set of all extremal non-negative martingales N = (N n = H(n, Z n ), n ∈ N). Considering only the case Z 0 = 1, then Remark 1.4 implies that the functions H are only defined for (n, k) such that k = r n 0 (mod L 0 ), where (L 0 , r 0 ) is the type of p. Let H denote the set of non-negative space-time function H such that there exists a sequence (a n , n ∈ N * ) with H(h, k) = lim n→∞ P k (Z n−h = a n )/P(Z n = a n ) for all h, k ∈ N. According to Kemeny, Snell and Knapp [23] Chapter 10, we have M ⊂ H.
Consider the collection H * = {H θ , θ ∈ [0, ∞)}. We deduce from Section 1.2 that H * ⊂ H. This appears already in Athreya and Ney [7], see also Section II.9 from Athreya and Ney [8]. We also deduce from Section 1.2 that H 0,0 ∈ H if and only if a = 0. We get a complete description of H and M in the Harris case and geometric case. To our knowledge, the results for the Harris case in the present work and for the geometric case in [1] are the first complete descriptions of the Martin boundary for super-critical GW process. This (partially) answers a question raised in [7], on the identification of H\H * . Theorem 1.9. Let p be a non-degenerate super-critical offspring distribution with finite mean. If b < ∞, then we have: under the L log(L) condition and the aperiodic condition, that is L 0 = 1. The L log(L) condition is satisfied in the Harris case (b < ∞) and in the geometric case. The periodic case is an immediate extension.) This result is based on the fact that for all θ ∈ (0, +∞) a.s. lim n→∞ z n (τ θ )/c n = θ. See Remark 5.3 for a slightly weaker result. The fact that H 0,0 (when it is defined) and H ∞ are extremal is immediate. Since M ⊂ H according to [23] Chapter 10, we deduce that M = H.
In the same spirit, Overbeck [30] has given an explicit description of the Martin boundary for some time-continuous branching processes, see for example Theorem 2 therein.
We conjecture that H = H * or H = H * ∪ {H 0,0 } as soon as b = +∞ and f (R c ) = +∞, keeping H 0,0 if and only if a = 0. Otherwise, existence of a limit function H when lim n→∞ a n /c n = +∞ is still open in the general case.
We end this section with some works related to Martin boundary for GW process. We refer to Dynkin [15] or to [23] for a presentation of the Martin boundary. For the extremal non-negative harmonic functions (space only) of GW process, we refer to Theorem 3 in Cohn [11], which is stated under the L log(L) condition and an aperiodic condition. (Notice that the L log(L) and aperiodic conditions are indeed required in the proof of Theorem 3 in [11] as it relies on Corollary 2.3.II a) from [26].) For the Martin entrance boundary of GW process, see Alsmeyer and Rösler [5].

The geometric offspring distribution case
We consider the geometric super-critical offspring distribution. We collect results developed in [1] and in this paper.
The family (τ θ , θ ∈ [0, +∞]) is continuous in distribution for local convergence. The random tree τ ∞ has only one node of infinite degree which happens to be the root. The

Organization of the paper
We recall the definition of trees, the local convergence and the distribution of the Galton-Watson tree τ in Section 2. Section 3 is devoted to the Kesten tree associated with τ . We introduce in Section 4 a probability distribution ρ θ,r in (4.5) which plays a crucial role to describe the local limits in the moderate regime. We present the local limits in the moderate regime in Section 5. The statements of the local convergence are in Section 6. The continuity of the local limits is studied in Section 7 and the partial results on the continuity at θ = +∞ are presented in Section 8. Section 9 is devoted to the sub-critical case (when it is seen as the super-critical case conditioned to the extinction event). After some ancillary results given in Section 10, we give detailed proofs in the technical Section 11 for the Harris case and state the results for the Böttcher case in Section 12.
We say that a function g defined on (0, +∞) is multiplicatively periodic with period c > 0 if g(cx) = g(x) for all x > 0. Notice that g is also multiplicatively periodic with period 1/c.

The set of discrete trees
We recall Neveu's formalism [28] for ordered rooted trees. Let U = n≥0 (N * ) n be the set of finite sequences of positive integers with the convention (N * ) 0 = {∂}. We also set U * = n≥1 (N * ) n = U\{∂}.
For u ∈ U, let |u| be the length or the generation of u defined as the integer n such that u ∈ (N * ) n . If u and v are two sequences of U, we denote by uv the concatenation of two sequences, with the convention that uv = vu = u if v = ∂. The set of strict ancestors of u ∈ U * is defined by Anc(u) = {v ∈ U, ∃w ∈ U * , u = vw}, and for S ⊂ U * , being non-empty, we set Anc(S ) = u∈S Anc(u).
A tree t is a subset of U that satisfies: We denote by T ∞ the set of trees. For r ∈N, r ≥ 1, we denote by t r the regular r-ary tree, defined by k u (t r ) = r for all u ∈ t r . Let t ∈ T ∞ be a tree. The vertex ∂ is called the root of the tree t and we denote by t * = t\{∂} the tree without its root. For a vertex EJP 24 (2019), paper 15. u ∈ t, the integer k u (t) represents the number of offspring (also called the out-degree) of the vertex u ∈ t. By convention, we shall write k u (t) = −1 if u ∈ t. The height H(t) of the tree t is defined by: For n ∈ N, the size of the n-th generation of t is defined by: We denote by T * f the subset of trees with finite out-degrees except the root's: and by T f = {t ∈ T * f ; k ∂ (t) < +∞} the subset of trees with finite out-degrees. Let h, k ∈ N * . We define T (h) f the subset of finite trees with height h: f ; k ∂ (t) = k} the subset of finite trees with height h and out-degree of the root equal to k. The restriction operators r h and r h,k are defined, for every t ∈ T ∞ , by:

Convergence of trees
Set N 1 = {−1}∪N, endowed with the usual topology of the one-point compactification of the discrete space {−1} ∪ N. For a tree t ∈ T ∞ , recall that by convention the outdegree k u (t) of u is set to -1 if u does not belong to t. Thus a tree t ∈ T ∞ is uniquely determined by the N 1 -valued sequence (k u (t), u ∈ U) and then T ∞ is a subset of N U

. By
Tychonov's theorem, the set N U 1 endowed with the product topology is compact. Since T ∞ is closed it is thus compact. In fact, the set T ∞ is a Polish space (but we don't need any precise metric at this point). The local convergence of sequences of trees is then characterized as follows. Let (t n , n ∈ N) and t be trees in T ∞ . We say that lim n→∞ t n = t if and only if lim n→∞ k u (t n ) = k u (t) for all u ∈ U. It is easy to see that: • If (t n , n ∈ N) and t are trees in T f , then we have lim n→∞ t n = t if and only if lim n→∞ r h (t n ) = r h (t) for all h ∈ N * . • If (t n , n ∈ N) and t are trees in T * f , then we have lim n→∞ t n = t if and only if lim n→∞ r h,k (t n ) = r h,k (t) for all h, k ∈ N * . If T is a T f -valued (resp. T * f -valued) random variable, then its distribution is charac- Using the Portmanteau theorem, we deduce the following characterization of convergence in distribution: • Let (T n , n ∈ N) and T be T f -valued random variables. Then, if a.s. H(T ) = +∞, we have: • Let (T n , n ∈ N) and T be T * f -valued random variables. Then, if a.s. H(T ) = +∞ and k ∂ (T ) = +∞, we have:

Galton-Watson trees
Let p = (p(n), n ∈ N) be a probability distribution on N. A T f -valued random variable τ is called a GW tree with offspring distribution p if for all h ∈ N * and t ∈ T f with H(t) ≤ h: The generation size process defined by (Z n = z n (τ ), n ∈ N) is the so-called GW process.
We refer to [8] and [6] for a general study of GW processes. We recall here the classical result on the extinction probability of the GW tree and introduce some notations. We denote by E = {H(τ ) < +∞} = n∈N {Z n = 0} the extinction event and denote by c the extinction probability: Then, if f denotes the generating function of p, c is the smallest non-negative root of f (s) = s. We denote by µ the mean of p i.e. µ = f (1). We recall the three following cases: • The sub-critical case (µ < 1): c = 1.
• The super-critical case (µ > 1): c ∈ [0, 1), the process has a positive probability of non-extinction. Notice that c = 0 if and only if a ≥ 1.
We consider the lower and upper bounds of the support of p: a = inf{n ∈ N; p(n) > 0} and b = sup{k; p(k) > 0} ∈N.
We say that p is non-degenerate if a < b. We define f n the n-th iterate of f , which is the generating function of Z n . We recall that lim n→∞ f n (0) = c. We also introduce in the supercritical case (µ > 1) the Schröder constant α defined by: (2.5) We set P k the probability under which the GW process (Z n , n ≥ 0) starts with Z 0 = k individuals and write P for P 1 so that: where the (Z (i) , 1 ≤ i ≤ k) are independent random variables distributed as Z under P. We consider a sequence (a n , n ∈ N * ) of elements of N and, when P(Z n = a n ) > 0, τ n a random tree distributed as the GW tree τ conditioned on {Z n = a n }. Let n ≥ h ≥ 1 and t ∈ T P(r h (τ n ) = t) = P(r h (τ ) = t) P k (Z n−h = a n ) P(Z n = a n ) · (2.6) EJP 24 (2019), paper 15.

The Kesten tree
In this section, we consider a GW tree τ with offspring distribution p = (p(n), n ∈ N) having mean µ ∈ (0, +∞). Recall that c ∈ [0, 1] denotes the extinction probability of τ . We define an associated probability distribution p on N as follows: Definition 3.1. (i) If c = 0, we define p as the Dirac mass at point a.
(ii) If c > 0, we define the probability distribution p = (p(n), n ∈ N) by: We denote by m the mean of p. If µ ≤ 1 and p(1) = 1, as c = 1, we have p = p and m = µ. If c > 0, we have m = f (c) ∈ (0, 1]. Remark 3.2. If c > 0, let τ 0,0 be a GW tree with offspring distribution p defined in (3.1). It is well known that the GW tree τ conditioned on the extinction event E is distributed as τ 0,0 . Indeed, we have using the branching property that, for h ∈ N * , t ∈ T (h) f , and setting k = z h (t): Let k ∈ N * . If f (k) (1) ∈ (0, +∞), that is p has finite moment of order k and the support of p is not a subset of {0, . . . , k − 1}, then we define the k-th order size-biased probability distribution of p as p [k] = (p [k] (n), n ∈ N) with: The generating function of (1). The probability distribution p [1] is the so-called size-biased probability distribution of p.
We now define the so-called Kesten treeτ 0 associated with the offspring distribution p. -The root is of type s.
-The number of offspring of a vertex depends, conditionally on the vertices of lower or same height, only on its own type (branching property).
-A vertex of type e produces only vertices of type e with offspring distribution p. -The random number of children of a vertex of type s has the size-biased distribution of p that is p [1] defined by (3.2) with k = 1. (Notice that p [1] is well defined as c > 0.) Furthermore, all of the children are of type e but one, uniformly chosen at random which is of type s.
(ii) If c = 0, the (degenerate) Kesten treeτ 0 is given by t a the regular a-ary tree, with a ≥ 1 defined by (2.4). It can be seen as a GW tree with degenerate offspring distribution the Dirac mass at point a. In this case all the individuals have type s.
Informally, when c > 0, the individuals of type s inτ 0 form an infinite spine on which are grafted independent GW trees distributed (see Remark 3.2) as τ conditionally on the extinction event E.
We define τ 0 = Ske(τ 0 ) as the treeτ 0 when one forgets the types of the vertices. If c = 0, then τ 0 is the regular a-ary tree. If c > 0, the distribution of τ 0 is given in the following classical result.
If µ ≤ 1, this is the usual link between Kesten tree and the size-biased GW tree. If µ > 1, the lemma just means that the Kesten tree is the sized biased tree associated with the tree conditioned on extinction (which is the subcritical GW tree with offspring distribution p). We give a short proof of this well-known result.
Proof. According to Section 2.2, the distribution of τ 0 is characterized by (3.3) for all h ∈ N * and t ∈ T and v ∈ t such that |v| = h. Let V be the vertex of type s at level h inτ 0 . We have, with k = z h (t): where we used (3.2) (with k = 1, n = k u (t) and p replaced by p) and (3.1) (with n = k u (t)) for the second equality and that u∈r h−1 (t) (k u (t) − 1) = k − 1 for the last one. Summing over all v ∈ t such that |v| = h gives the result.

A distribution associated with super-critical GW trees
In this section, we consider a super-critical GW tree τ with non-degenerate offspring distribution p = (p(n), n ∈ N) with finite mean µ ∈ (1, +∞). We recall that f denotes the generating function of p and c is the smallest root in [0, 1) of f (s) = s. Notice that a = 0 is equivalent to c > 0.
Following [33] or [6], we consider the Seneta-Heyde norming: (c n , n ∈ N) is a sequence such that e −Zn/cn , n ∈ N is a martingale and c 0 ∈ (−1/ log(c), +∞). This sequence is increasing positive and unbounded. Furthermore, we have that a < c n+1 /c n < µ for all n ∈ N and that the sequence (c n+1 /c n , n ∈ N) is increasing 2 and converges towards µ. We also have that (Z n /c n , n ∈ N) converges a.s. towards a non-negative random variable W with Laplace transform ϕ(λ) = E e −λW such that ϕ(+∞) = P(W = 0) = c and for all λ ≥ 0: f (ϕ(λ/µ)) = ϕ(λ). The probability distribution of W , up to a multiplicative constant, is the unique probability distribution solution of (4.1).

Remark 4.1.
If one assumes that p satisfies E[Z 1 log(Z 1 )] < +∞, then Kesten and Stigum results asserts that (µ −n Z n , n ∈ N) converges a.s. towards W up to a scaling factor and that lim n→∞ µ −n c n exists and belongs to (0, +∞).
We end this section with the limit of ρ θ,r as θ goes to 0 and in a particular case to +∞. Recall Definitions (2.4) and (2.5). One has to distinguish two cases when θ goes to 0: the so-called Schröder case a ≤ 1 (equivalently p(0) + p(1) = 0, f (c) > 0 or α < +∞) and the so-called Böttcher case a ≥ 2 (equivalently p(0) + p(1) = 0, f (c) = 0 or α = +∞). When θ goes to infinity we consider the particular so-called Harris case where p has a finite support (equivalently b is finite). (i) In the Schröder case (a ≤ 1), we get that ρ θ,1 converges to the Dirac mass at point 1 as θ goes down to 0. (ii) In the Böttcher case (a ≥ 2), we get that, for all r ∈ N * , ρ θ,r converges to the Dirac mass at (a, . . . , a) ∈ N r as θ goes down to 0. (iii) In the Harris case (b < ∞), we get that, for all r ∈ N * , ρ θ,r converges to the Dirac mass at (b, . . . , b) ∈ N r as θ goes to infinity.
Proof. We give the proof of (i). The technical proofs of (ii) and (iii) are postponed respectively to Sections 12.3 and 11.3. According to [10], there exists a positive continuous multiplicatively periodic function V defined on (0, +∞) with period µ such that for all x > 0: We have for θ > 0 as θ goes down to 0: where we used Definition (2.5) of the Schröder constant for the first equality and that V has multiplicative period µ for the last one. This implies that lim θ→0 ρ θ,1 (1) = 1 and thus ρ θ,1 converges to the Dirac mass at 1 as θ goes down to 0.

Extremal GW trees
We are in the setting of Section 4. If c > 0, we define the sub-critical offspring distribution p by (3.1) and, see (3.2), the corresponding size-biased distribution p [ ] of order ∈ N * . For ∈ N * such that f ( ) (c) > 0, we have: Let θ ∈ (0, +∞). We define a two-type random treeτ θ and shall consider the corresponding tree τ θ = Ske(τ θ ) when one forgets the types of the vertices ofτ θ . EJP 24 (2019), paper 15. Definition 5.1 (Extremal tree). Let p be a non-degenerate super-critical offspring distribution with finite mean. The labeled random treeτ θ is a two-type random tree where the vertices are either of type s (for survivor) or of type e (for extinction) and τ θ = Ske(τ θ ) denotes the corresponding random T f -valued tree when one forgets the labels (or types).
The distribution ofτ θ is characterized as follows: • The root is of type s. • The number of offspring of a vertex of type e does not depend on the vertices of lower or same height (branching property for vertices of type e). • A vertex of type e produces only vertices of type e with offspring distribution p (as in the Kesten tree).
For a vertex u of type s, we denote by κ s (u) the number of children of u with type s and by κ e (u) the number of children of u with type e. Conditionally given r h (τ θ ) and (S , 0 ≤ ≤ h), we have: and the s u vertices of type s are chosen uniformly at random among the k u (τ θ ) children.
Notice that Property (i) in the above definition breaks down the branching property.
If c = 0, then a.s. κ e (u) = 0, so that there are no individuals of type e. We stress, and shall use later on, thatτ θ truncated at level h can be recovered from r h (τ θ ) and S h as all the ancestors of a vertex of type s are of type s and a vertex of type s has at least one child of type s. Since all the vertices of type s have at least one offspring of type s, we get S h+1 ≥ S h . The offspring distribution of vertices of type s can also be described as follows. For every h ≥ 0, conditionally given r h (τ θ ) and S h , we compute the probability that • we have S h+1 − S h = n for some n ≥ 0 i.e. n new vertices of type s appear at generation h + 1, • every node u of S h has k u offspring, s u of them being of type s, where the integers ((s u , k u ), u ∈ S h ) satisfy 1 ≤ s u ≤ k u and u∈S h s u = n + S h , • for every u ∈ S h and every subset A u ⊂ {1, . . . , k u } such that A u = s u , the positions of the offspring of u of type s among all the offspring of u, are given by A u i.e. S h+1 ∩ {u1, . . . , uk u } = uA u where we recall that uv denotes the concatenation of the two sequences u and v.
We have: where we used (4.5) and (5.1) for the last equality.
By construction, a.s. individuals of type s have a progeny which does not suffer extinction whereas individuals of type e (if any) have a progeny which suffers extinction. Since the individuals of type s do not satisfy the branching property, the random treeτ θ is not a two-type inhomogeneous GW tree.
Using this definition, it is easy to get that the distribution of the tree r h (τ θ ) is absolutely continuous with respect to those of the original GW tree r h (τ ). Lemma 5.2. Let p be a non-degenerate super-critical offspring distribution with finite mean. Let θ ∈ (0, +∞). Let h ∈ N * and t ∈ T In order to shorten the notations, we set A = S h Anc(S h ). We set, for ∈ {0, . . . , h − 1}, S = {u ∈ A, |u| = } the vertices at level which have at least one descendant in S h . For u ∈ r h−1 (t), we set s u (t) = (A uN * ), the number of children of u having descendants in S h . We recall thatτ θ truncated at level h can be recovered from r h (τ θ ) and S h . We where we used that for a tree s, we have u∈r h−1 (s) k u (s) − 1 = z h (s) − 1 and that s = A is tree-like with z h (s) = S h . Remark that C S h depends only of S h . Since S h ≥ 1 as the root is of type s, we obtain: where we used (4.4) for the last equality.
f , and g a non-negative measurable function defined on R + , that: This implies that for every non-negative measurable function G defined on T ∞ × R + , we have: Thus, the distribution probability of τ θ is a regular version of the distribution of τ conditionally on {W = θ}. From Lemma 5.2, we get that this version is continuous on T (h) f for all h ∈ N * . In particular, we deduce that for a.e. θ ∈ (0, +∞), a.s. lim n→∞ z n (τ θ )/c n = θ (see also Theorem 2.II in [26] for an a.s. convergence for all θ ∈ (0, +∞) under stronger EJP 24 (2019), paper 15. hypothesis). The distribution of τ conditionally on E c can be written as a mixture of distributions of τ θ as for every Borel set A of T ∞ ,

Convergence of conditioned super-critical GW trees
We are in the setting of Section 4, with τ a GW tree with super-critical non-degenerate offspring distribution p with finite mean µ. We consider a deterministic N-valued sequence (a n , n ∈ N * ) such that P(Z n = a n ) > 0 for every n > 0. See Remark 1.4 for conditions on the existence of such sequences. We denote by τ n a random tree distributed as the GW tree τ conditioned on {Z n = a n }. We study the limit in distribution of τ n as n goes to infinity and we consider different regimes according to the growth speed of the sequence (a n , n ∈ N * ). Recall that Z n is under P k distributed as a GW process with offspring distribution p starting at Z 0 = k.
We say that the offspring distribution p is of type (L 0 , r 0 ), when L 0 is the period of p, that is the greatest common divisor of {n − ; n > and p(n)p( ) = 0}, and r 0 is the residue (mod L 0 ) of any n such that p(n) = 0. See Remark 1.4 on sufficient conditions to get P k (Z n = a) > 0.

The intermediate regime:
lim n→∞ a n /c n ∈ (0, +∞) We first state a strong ratio limit which is a direct consequence of the local limit theorem in [12]. Lemma 6.1. Let p be a non-degenerate super-critical offspring distribution with finite mean and type (L 0 , r 0 ). Let θ ∈ (0, +∞). Assume that lim n→∞ a n /c n = θ and that a n = r n 0 (mod L 0 ) for all n ∈ N * . For all h, k ∈ N * , we have: Proof. The local limit theorem in [12] states that for all k ∈ N * , θ ∈ (0, +∞) and (a n , n ∈ N) a sequence of elements of N * such that lim n→∞ a n /c n = θ, we have: We now assume that a n = kr n 0 (mod L 0 ) and lim n→∞ a n /c n = θ ∈ (0, +∞). Using Remark 1.4, we deduce that P k (Z n−h = a n ) > 0 if and only if a n = kr n−h 0 (mod L 0 ) that is k = r h 0 (mod L 0 ). In this case, noticing that lim n→∞ a n /c n−h = µ h θ as lim n→∞ c n /c n−h = µ h , using (6.1), we get that: We deduce the following local convergence.
Proposition 6.2. Let p be a non-degenerate super-critical offspring distribution with finite mean. Let θ ∈ (0, +∞). Assume that lim n→∞ a n /c n = θ and that τ n is well defined for all n. Then, we have the following convergence in distribution: Proof. Assume that p is of type (L 0 , r 0 ), so that τ n is well defined for n large if and only if a n = r n 0 (mod L 0 ). Using that a.s. H(τ θ ) = +∞, the characterization (2.1) of the convergence in T f , (2.6) with k = r h 0 (mod L 0 ), and Lemmas 5.2 and 6.1, we directly get the result.

6.2
The high regime in the Harris case: lim n→∞ a n /c n = +∞ Let p be a non-degenerate super-critical offspring distribution with finite mean. Recall b (the supremum of the support of p) defined in (2.4). Notice that b finite (Harris case) implies that p has finite mean. When b < ∞, we define τ ∞ as t b , the deterministic regular b-ary tree. Proposition 6.3. Let p be a non-degenerate super-critical offspring distribution with b < ∞. Assume that a n ≤ b n for all n ∈ N * , lim n→∞ a n /c n = ∞ and that τ n is well defined for all n. Then, we have the following convergence in distribution: Proof. We assume that τ n is well defined, that is P(Z n = a n ) > 0. For h ∈ N * , we have . We deduce from (2.6) and (2.1), using that t b has a.s. an infinite height, that the proof of Proposition 6.3 is complete as soon as we prove It is also enough to consider the two cases: lim n→∞ a n /b n = 1 or lim sup n→∞ a n /b n < 1 with lim n→∞ a n /c n = +∞. We first consider the case lim n→∞ a n /b n = 1. Notice that P k (Z n−h = a n ) = 0 for kb n−h < a n as each individual produces at most b children.
Since lim n→∞ a n /b n = 1, we deduce that for h, k ∈ N * , if k ≤ b h − 1, then kb n−h < a n for n large enough. Using (6.3), we deduce that for n large enough, P(Z n = a n ) = P(Z h = b h )P b h (Z n−h = a n ) as soon as P(Z n = a n ) > 0. This gives (6.2).
The case lim sup n→∞ a n /b n < 1 and lim n→∞ a n /c n = +∞ is proven in Section 11.4, see Lemma 11.6 with = 1.
6.3 The low regime: lim n→∞ a n /c n = 0 Let p be a non-degenerate super-critical offspring distribution with finite mean. If c > 0 (and thus a = 0), we recall that τ 0,0 denote the distribution of the GW tree τ with offspring distribution p given in (3.1). According to Remark 3.2, we have the following result for the extinction regime. Proposition 6.4. Let p be a non-degenerate super-critical offspring distribution with finite mean such that c > 0. Assume that a n = 0 for n large enough so that τ n is well defined for n large enough. Then, we have the following convergence in distribution: Recall the Kesten tree τ 0 from Definition 3.3. Recall that a ≥ 1 implies that a.s. τ 0 = t a , the deterministic regular a-ary tree. Proposition 6.5. Let p be a non-degenerate super-critical offspring distribution with finite mean. Assume that a n ≥ 1 ∨ a n for all n ∈ N * , lim n→∞ a n /c n = 0 and that τ n is well defined for all n. Then, we have the following convergence in distribution: Proof. We give the proof in the Schröder case (a ≤ 1). The Böttcher case (a ≥ 2) is more technical and its proof is postponed to Section 12.5. We suppose throughout the proof that p is of type (L 0 , r 0 ).
Case I: the sequence (a n , n ∈ N * ) is bounded. We first consider the case a = 0. The ratio theorem, see (4) in [31] (or [8] Theorem A.7.4), implies, that for all , k, h ∈ N * , if P(Z n = k) > 0 for n large enough, then: We deduce from (2.6) and (3.3), Since τ 0 has a.s. an infinite height, we get that τ n converges in distribution towards τ 0 using the convergence characterization (2.1).
We consider now the case a = 1. Recall that t a is the regular a-ary tree. According to Remark 1.4, for k large enough, we get that P(Z n = k) > 0 and P(Z n−h = k) > 0 for n large enough. It is easy to check that for h ∈ N, k ∈ N * : For k = 1, the left hand side member is equal to one. For k > 1, it is not difficult to get, by considering the lowest vertex of τ with out-degree larger than one, that the sequence (P(Z n = k)/P(Z n = 1), n ∈ N * ) is bounded. Then arguing as in [31], one gets that lim n→∞ This implies that τ n converges in distribution towards τ 0 = t a using the convergence characterization (2.1).
Case II: lim n→∞ a n = +∞. We first consider the case a = 0. Then we have f n (0) > 0 for we deduce from Lemma 6.6, stated below, that: Since τ 0 has a.s. an infinite height, we deduce that (6.4) holds for all t ∈ T Since singletons are open subsets of the closed discrete set 0≤h ≤h T (h ) f , we deduce from the Portmanteau theorem that (r h (τ n ), n ∈ N) converges in distribution towards r h (τ 0 ). Since this holds for all h ∈ N * , and since τ 0 has a.s. an infinite height, we conclude using the convergence characterization (2.1).
We now consider the case a = 1. Then we have a.s. τ 0 = t a . We deduce, as f (c) = p(1), that P(r h (τ ) = r h (t a )) = f (c) h and thus, using (2.6) and Lemma 6.6: Since this holds for all h ∈ N * , and since t a has a.s. an infinite height, we conclude using the convergence characterization (2.1).
The proof of the previous proposition in the Schröder case is based on the following strong ratio limit. Lemma 6.6. Let p be a non-degenerate super-critical offspring distribution with finite mean in the Schröder case (a ≤ 1). Assume that lim n→+∞ a n = +∞, lim n→+∞ a n /c n = 0 and P(Z n = a n ) > 0 for every n ∈ N * . Then we have for all h ∈ N * : Notice that according to Remark 1.4, the condition P(Z n = a n ) > 0 in Lemma 6.6 is satisfied as soon as a n = r n 0 (mod L 0 ) as lim n→+∞ a n = +∞ and lim n→+∞ a n /c n = 0.
Proof. Since a ≤ 1, we have r 0 ∈ {0, 1}. We deduce from Corollary 5 in [17], that for k n ≤ c n and lim n→∞ k n = +∞: The hypothesis on a n imply thus that lim n→∞ ρ an = +∞. Set ρ = ρ an for simplicity. Assume that a n = r n 0 (mod L 0 ), so that P(Z n = a n ) > 0 for n large enough. For n large enough, we have: where we used (6.6) for the first approximation, the representation (4.7) of w in the Schröder case and that V is multiplicatively periodic with period µ for the second one.
7 Continuity in law of the extremal GW trees at θ = 0 We are in the setting of Section 4. Recall the definition ofτ θ given in Section 5 for θ > 0 and in Section 3 for θ = 0. Since the function w is continuous, we get that the distribution ofτ θ and thus of τ θ , as a function of θ ∈ (0, +∞) is continuous. From the convergence of the offspring distribution of the individuals of type s which is a consequence of Lemma 4.4, we deduce the continuity in distribution ofτ θ for θ ∈ [0, +∞). This directly gives the continuity in distribution of τ θ for θ ∈ [0, +∞). We stress in the next corollary that only the convergence at 0 is non-trivial.
As a consequence of Corollary 7.1, we recover directly Corollary 3 from [9], which is stated only in the Böttcher case (p(0) + p(1) = 0) and extend it to the Schröder case, see next corollary. Recall that in the Böttcher case, the random tree τ 0 is in fact the (deterministic) regular a-ary tree. For ε ∈ (0, 1), let τ (ε) be distributed as τ conditionally on {0 < W ≤ ε}. Notice that if c = 0, then conditioning on {0 < W ≤ ε} is the same as conditioning on {0 ≤ W ≤ ε}. Proof. Let h ∈ N * and t ∈ T (h) f and set k = z h (t). We deduce from Lemma 5.2 that for all θ ∈ (0, +∞): Integrating with respect to θ ∈ (0, ε] for some ε > 0, we get: are independent random variables distributed as W under P. Using Corollary 7.1, we get that: This implies that: On the other hand, we have: where we used that lim n→∞ c n /c n+h = µ −h for the last equality. We deduce that: Then use (7.1) and the characterization (2.1) of the convergence in T f to conclude.

Weak continuity in law of the extremal GW trees at θ = +∞
We are in the setting of Section 4. We introduce here the tree τ ∞ presented in Definition 1.1 that will be the good candidate for the limiting tree of the conditioned GW tree in the high regime.
In Section 8.1, we introduce a whole family (T (λ) ) λ≤λc of inhomogeneous GW trees which converges in distribution to τ ∞ as λ → λ c . These trees are first constructed by absolute continuity with respect to the distribution of r h (τ ) and in Section 8.2 viewed as finite trees grafted on an infinite backbone of immortal particles. This two type GW trees generalize the Kesten tree and the trees τ θ of Section 5.
In Section 8.3, the tree τ ∞ is indeed proven to be the limit in distribution of the trees τ θ as θ → +∞ for the geometric offspring distribution, see [1], and the Harris case, see Proposition 8.10. In general, this convergence is more involved and we prove only a weak limit in Proposition 8.7 which implies that τ ∞ is the only possible limit, if any, for τ θ as θ → +∞, see Corollary 8.8, but the proof of the convergence remains open.
We give in Section 8.4 an alternative description of the tree T (λ) as a two type GW tree, where trees distributed as τ (which are thus possibly infinite) are grafted on an infinite backbone.

A family of two-type GW trees
We keep notations from Section 8.1. For λ ∈ (−∞, λ c ], we give a description of T (λ) using a two-type GW treeT (λ),e .
For h ∈ N and λ ∈ (−∞, λ c ] such that ζ h (λ) is finite, we define the probability Notice thatp We define a two type random treeT (λ),e in the next definition and write T (λ),e = Ske(T (λ),e ) for the treeT (λ),e when one forgets the types of the vertices ofT (λ),e . EJP 24 (2019), paper 15. Definition 8.3. Let p be a non-degenerate super-critical offspring distribution with finite mean. Let λ ∈ (−∞, λ c ]. The labeled treeT (λ),e is a two-type random tree whose vertices are either of type s (for survivor) or of type e (for extinct).
(i) If ζ 0 (λ) = ζ 1 (λ) = +∞, thenT (λ),e is the regular b-ary tree and all its vertices are of type s (and thus there is no vertex of type e).
(ii) If ζ 1 (λ) < +∞, the random treeT (λ),e is defined as follows: -For a vertex, the number of offspring of each type and their positions depend, conditionally on the vertices of lower or same height, only on its own type (branching property).
-The root is of type s with probability (ζ 0 (λ) − c)/ζ 0 (λ). This probability is set to 1 if ζ 0 (λ) = +∞. -A vertex of type e produces only vertices of type e with sub-critical offspring distribution p.
-If ζ 0 (λ) = +∞, then the root, which is of type s a.s., has infinitely many children of types s and e, each children being, independently from the other, of type s with probability (ζ 1 (λ) − c)/ζ 1 (λ). That is for k 0 ∈ N * and S 1 ⊂ {1, . . . , k 0 }: . Unless a ≥ 1 or ζ 0 (λ) = ζ 1 (λ) = +∞, conditionally on the fact that the root is of type s, a.s. there exists an infinite number of vertices of type s and of type e. By construction individuals of type s have a progeny which does not suffer extinction, whereas individuals of type e have a finite progeny. Informally the individuals of type s inT (λ),e , if any, form a backbone, on which are grafted, if a = 0, independent GW trees distributed as τ conditionally on the extinction event E. This is in a sense a generalization of the Kesten tree, where the backbone is reduced to an infinite spine in the case a ≤ 1. We stress that T (λ),e , truncated at level h can be recovered from r h (T (λ),e ) and S h as all the ancestors of a vertex of type s is also of a type s and a vertex of type s has at least one children of type s.
The following result states that the random tree T (λ) can be seen as the skeleton of a two-type GW tree.  Proof. Let λ ∈ (−∞, λ c ]. We first consider the case ζ 0 (λ) finite. We assume c > 0 (or equivalently a = 0). Let h ∈ N * , t ∈ T (h) f and S h ⊂ {u ∈ t; |u| = h}. Set k = z h (t) = {u ∈ t; |u| = h}. In order to shorten the notations, we set A = S h Anc(S h ). We set, for ∈ {0, . . . , h − 1}, S = {u ∈ A, |u| = } the vertices at level which have at least one descendant in S h . For u ∈ r h−1 (t), we set s u (t) = (A uN * ), the number of children of u having descendants in S h . We recall thatT (λ),e truncated at level h can be recovered from r h (T (θ),e ) and S h . We compute C S h = P(r h (T (λ),e ) = t, S h = S h ). We have by construction if S h > 0: where we used that for a tree s, we have u∈r h−1 (s) k u (s) − 1 = z h (s) − 1 and that s = A is tree-like with z h (s) = S h . It is elementary to check that Formula (8.10) is also true when S h is empty, and the root is thus of type e. Since C S h depends only of S h , we shall write C S h for C S h . We get: We deduce from (8.7) that T (λ),e and T (λ) have the same distribution. The case ζ 0 (λ) finite and c = 0 (i.e. a > 0) is clear, as there is no vertex of type e in T (λ),e and the offspring distribution of individuals of type s at level h inT (λ),e given by (8.9), that is:p coincides with the offspring distributionp We consider the case ζ 0 (λ) = +∞, ζ 1 (λ) finite and c > 0. Let k 0 , h ∈ N * , t ∈ T (h) k0 and S h ⊂ {u ∈ t; |u| = h}. Set k = z h (t) = {u ∈ t; |u| = h}. Arguing as in the case ζ 0 (λ) finite, we get if c > 0: and thus, writing C S h for C S h as the latter quantity depends only on S h : Then use (8.8) to conclude. The sub-case c = 0 is handled in the same way as when ζ 0 (λ) is finite.
Eventually, we consider the case ζ 1 = +∞. In this case T (λ),e and T (λ) are by definition regular b-ary trees, and they are thus a.s. equal.
For λ > −∞, we denote by T (λ), * the tree-valued random variable distributed as T (λ) conditionally on the non extinction event (which is distributed as the skeleton ofT (λ),e conditionally on the root being of type s). Recall the Kesten tree τ 0 defined in Section 3.
Proof. Considering the cases a = 0 and a ≥ 1, it is easy to check that the distributionŝ p (λ),e h overN defined in (8.9) converge as λ goes to −∞ towards the Dirac mass at max(1, a). This implies the convergence in distribution as λ goes to −∞ ofT (λ),e conditionally on the root being of type s towardsτ 0 . Using that the extinction event of T (λ),e corresponds to the root ofT (λ),e being of type s, we obtain the convergence of the lemma.
Remark 8.6. In the proof of Lemma 8.5, we proved in fact the convergence of the two-type random treesT (λ),e conditionally on the root being of type s towardsτ 0 as λ goes to −∞, using the convergence in distribution of the probability distributionp (λ),e as λ goes to −∞.

Continuity in law of the extremal GW trees at θ = +∞
Recall that T (λ), * is distributed as T (λ) conditionally on the non extinction event (which is distributed as the skeleton ofT (λ),e conditionally on the root being of type s).
Proof. Let h ∈ N * , t ∈ T (h) f and S h ⊂ {u ∈ t; |u| = h} with S h non empty. We recall that the distribution ofτ θ up to generation h is completely characterized by r h (τ θ ) its skeleton up to level h and by the set S h of vertices at generation h which are of type s. We still denote by S h the vertices of τ Θ λ at generation h which are of type s. We have with k = z h (t) and = S h : EJP 24 (2019), paper 15.
where we used (5.3) for the second equality, that (W i , i ∈ N * ) are independent random variables distributed as W for the third one and the definition of ζ h given in (8.1). Then use (8.10) and that the root ofT (λ),e is of type s with probability (ζ 0 (λ) − c)/ζ 0 (λ) to get that: Sinceτ Θ λ up to level h is characterized by τ Θ λ and S h , and similarly forT (λ),e , we deduce from the previous equality thatτ Θ λ is distributed asT (λ),e conditionally on its root being of type s. Then, forgetting about the types, we deduce that τ Θ λ is distributed as T (λ), * .
When λ goes to −∞, we get that the measure g λ (θ) dθ converges weakly to the Dirac mass at 0. We deduce that Θ λ converges in distribution towards 0 as λ goes to −∞.
We then recover from Proposition 8.7 and Corollary 7.1 the convergence in distribution of T (λ), * , that is of T (λ),e conditionally on the non-extinction event, towards τ 0 given in Lemma 8.5.
If E[e λcW ] = +∞ (and thus λ c > 0) or equivalently f (R c ) = +∞, then when λ goes up to λ c we get that Θ λ converges in distribution towards +∞. We deduce from Lemma 8.1 the following corollary. Corollary 8.8. Let p be a non-degenerate super-critical offspring distribution whose generating function blows-up (that is f (R c ) = +∞). Then, if (τ θ , θ ∈ [0, ∞)) converges in distribution as θ goes to infinity, then the limit is the distribution of τ ∞ . Remark 8.9. If R c = +∞, then the tree τ ∞ has all its nodes with degree b ∈N. Since the distribution of τ ∞ is maximal in the convex set of probability distributions on T ∞ , we get that the distribution of τ ∞ is the limit in distribution of a sub-sequence (τ θn , n ∈ N) with lim n→∞ θ n = +∞.
We are able to prove the stronger result on the convergence in distribution of (τ θ , θ ∈ [0, ∞)) as θ goes to infinity in the particular case of the geometric offspring distribution (in this case λ c is positive finite, E[e λcW ] = +∞ and b = ∞), see [1]. The next proposition, which is a direct consequence of the convergence of ρ θ,r as θ → +∞ given in Lemma 4.4, asserts that it also holds if the offspring distribution has a finite support which is the so-called Harris case (in this case b < ∞ and λ c = +∞). Otherwise, the general case is open. Proposition 8.10. Let p be a non-degenerate super-critical offspring distribution with finite support, that is b < +∞ (Harris case). Then we have the following convergence in distribution:

A remark on an other trees family
We provide in this section an alternative description of T (λ) using a two-type GW treê T (λ),n .
We assume that λ c > 0. Notice that the sequence (ζ h (λ), h ∈ N) defined in (8.1) is non-increasing and ζ h (λ) > 1 for all h ∈ N, λ ∈ (0, λ c ]. Furthermore, as R c > 1, we get that f ( ) (1) is finite for all ∈ N. For h ∈ N and λ ∈ (0, λ c ] such that ζ h (λ) is finite, we define the probabilityp For ∈ N such that ≤ b, we recall the th-size biased probability distribution of p defined in (3.2). We define a two type random treeT (λ),n in the next definition and write T (λ),n = Ske(T (λ),n ) as the treeT (λ),n when one forgets the types of the vertices ofT (λ),n .
Definition 8.11. Let p be a non-degenerate super-critical offspring distribution such that λ c > 0. Let λ ∈ (0, λ c ]. We define a labeled random treeT (λ),n , whose vertices are either of type s (for survivor) or of type n (for normal).
(i) If ζ 0 (λ) = ζ 1 (λ) = +∞, thenT (λ),n is the regular b-ary tree and all its vertices are of type s (and thus there is no vertex of type n). (ii) If ζ 1 (λ) < +∞, the random treeT (λ),n is defined as follows: -For a vertex, the number of offspring of each type and their positions depend, conditionally on the vertices of lower or same height, only on its own type (branching property).
A vertex u ∈T (λ),n at level h of type s produces κ s (u) vertices of type s with probability distributionp (λ),n h and κ n (u) vertices of type n such that conditionally on κ s (u) = s u ≥ 1, k u (T (λ),n ) = κ s (u) + κ n (u) has distribution p [su] , defined in (3.2), and the s u individuals of type s are chosen uniformly at random among the k u (T (λ),n ) children. More precisely if we denote by S h = {u ∈ T (λ),n ; |u| = h and u is of type s} the set of vertices ofT (λ),n with type s at level h ∈ N, and we have for u ∈ S h : for all k u ∈ N * , s u ∈ {1, . . . , k u }, and A u ⊂ {1, . . . , k u } such that A u = s u , P κ s (u) + κ n (u) = k u and S h+1 ∩ {u1, . . . , If ζ 0 (λ) = +∞, then the root, which is of type s a.s., has infinitely many children of type s and n, each children being, independently from the other, of type s with probability (ζ 1 (λ) − 1)/ζ 1 (λ). That is for k 0 ∈ N * and S 1 ⊂ {1, . . . , k 0 }: The main difference withT (λ),e is that the individuals of type s inT (λ),n , if any, form a backbone on which are grafted, if a = 0, independent GW trees distributed as τ (instead of τ conditionally on the extinction event E inT (λ),e ).
The following result states that the random tree T (λ) can also be seen as the skeleton of this new two-type GW tree. Its proof, which follows the proof of Lemma 8.4, is left to the reader.

Remark 8.13.
Recall that ζ 0 (λ) > 1 for λ ∈ (0, λ c ]. For λ ∈ (0, λ c ], such that ζ 0 (λ) is finite, we consider the function h λ defined by: Since, by definition, h λ = 1, we deduce that h λ is a probability density. Let Θ λ be a random variable with density h λ . We consider the random tree τ Θ λ and the random two-type treeτ Θ λ , which conditionally on {Θ λ = θ} are distributed respectively as τ θ and τ θ . Computation similar as in (8.11) gives that for h ∈ N * and t ∈ T (h) f , with k = z h (t), and S h ⊂ {u ∈ t; |u| = h} with S h non empty and = S h : Similar computations as in (8.10) give that: Summing over all non-empty subsets S h of {u ∈ t; |u| = h}, gives that: Thus the random tree τ Θ λ is distributed as T (λ),n conditionally on the root being of type s.

Convergence of conditioned sub-critical GW tree
In this section, we consider a sub-critical GW tree τ with general non-degenerate offspring distribution p = (p(n), n ∈ N) with finite mean µ ∈ (0, 1). To avoid trivial cases, we assume that p(0) + p(1) < 1. We denote by f the generating function of p. We assume that there exists κ > 1 such that f (κ) = κ and f (κ) < +∞. Since f is strictly convex, κ, when it exists, is unique. Those assumptions are trivially satisfied if the radius of convergence of f is infinite. This is also the case for geometric offspring distribution studied in [1].
We have thatp defined by (3.1) (with p replaced byp) is equal to p by construction. Notice that we are in the Schröder case and that p is of type (L 0 , 0) asp(0) > 0. Letτ be the corresponding super-critical GW tree. It is elementary to check that for h ∈ N * and t ∈ T (h) f , we have with k = z h (t): P(r h (τ ) = t) = κ k−1 P(r h (τ ) = t). (9.1) Recall that Z n = z n (τ ), and setZ n = z n (τ ). Following Section 4, let (c n , n ∈ N) be a sequence with c 0 > 0 such that κ Zn e −Zn/cn , n ∈ N or equivalently e −Zn/cn , n ∈ N is a martingale. This sequence is increasing positive and unbounded. Furthermore, the sequence (c n+1 /c n , n ∈ N) increases towardsμ = f (κ).
We consider a sequence (a n , n ∈ N * ) of integers such that P(Z n = a n ) > 0 (see Remark 1.4). We denote by τ n (resp.τ n ) a GW tree distributed as τ (resp.τ ) conditionally on {Z n = a n } (resp. {Z n = a n }). Clearly if a n = 0 for n large enough, then (τ n , n ∈ N * ) converges in distribution towards τ . So only the case a n positive for n ∈ N * is of interest.
It is straightforward to deduce from (9.1) that for n ≥ h ≥ 1 and t ∈ T  denote byτ ∞ the deterministic regular b-ary tree. Letτ 0 be defined as the Kesten tree τ 0 in Definition 3.3 where p is equal to p. We deduce from Propositions 6.2, 6.5 and 6.3, (9.2) and the characterization (2.1) of the convergence in T f the following result.
Proposition 9.1. Let p be a non-degenerate sub-critical offspring distribution with generating function f such that b ≥ 2 and suppose that there exists (a unique) κ > 1 such that f (κ) = κ and f (κ) < +∞. Let θ ∈ [0, +∞). Assume that lim n→∞ a n /c n = θ, a n > 0 and τ n is well defined for all n ∈ N * . Then, we have the following convergence in distribution: If b is finite, then (9.3) holds also for θ = ∞.
In the sub-critical regime, the local convergence of τ n and the identification of the limit if any when 1 is the only root of the equation f (κ) = κ is an open question.

Ancillary results
We adapt the proof of Theorem 1 in [16]. Recall that W , conditionally on {W > 0} has a positive continuous density w on (0, +∞). We shall use the following well known result.
Lemma 10.1. Let X be a real random variable with a continuous density. Let a < b be elements of {λ ∈ R; E[e λX ] < +∞}. For z ∈ C such that R(z) ∈ K = [a, b], the Laplace transform g(z) = E[e zX ] is well defined and we have: Let t 0 > 0. There exists η ∈ (0, 1) such that for all u ∈ K, t ∈ R with |t| ≥ t 0 , we have: .

Lemma 10.2.
Let p be a non-degenerate super-critical offspring distribution with finite mean. Let a < 0 ≤ b such that K 0 := [a, b] ⊂ K. Let t 0 > 0. There exists η ∈ (0, 1) such that for all u ∈ K 0 , t ∈ R with |t| ≥ t 0 : This gives the result.
We now prove (10.10) in the Schröder case. There exists η ∈ (0, 1/2) such that c < (1 − 2η ) 2φ (a). We can choose an integer k 0 ≥ k 0 such that c + Bm k 0 < (1 − η ) 2φ (a). We can also choose n 0 ≥ k 0 large enough so that c n0 > c 0 µ k 0 and inf n≥n0φn (a) ≥ (1−η )φ(a). Notice that for n ≥ k 0 : Using (10.16), we get that for k ∈ N * , n ≥ n 0 with k + k 0 ≤ n, u ∈ K 0 , t ∈ J n,k : This gives that |φ n (u + it)| ≤ (1 − η )φ n (u) for all u ∈ K 0 , t ∈ πc0µ k 0 L0 , πcn We present detailed proofs of the results, because even if they correspond to an adaptation of the results known in the Böttcher case (see [17] and [18]), we believe that the adaptation is not straightforward since in particular the Fourier inversion of w * is not valid if α ≤ 1. We keep notations from Sections 2.3 and 4. Recall b defined in (2.4) is the supremum of the support of the offspring distribution p. We assume b < ∞ (Harris case). Following [16] or [10], we define the (right) Böttcher constant β H ∈ (1, +∞) by:
Lemma 11.2. For all s ∈ (1, +∞) and all n ∈ N * , we have: This gives the result.
Plugging this in (11.15) we get that:

Upper large deviations for Z n
Recall Definition (4.2) of K and notations from Section 10, and in particular Definition (10.3) of K . In the Harris case, we have K = K = R. We recall that for j ∈ N * , ϕ j (z) = E[e zWj ] = f j (e z/cj ), with W j = Z j /c j , is well defined for z ∈ C and thatφ j converges uniformly on the compacts of C towardsφ as j goes to infinity. Elementary computations give that lim u→+∞φ j (u)/φ j (u) = b j /c j .
We consider the functionsψ j =b •φ j defined on some open neighborhood of (0, +∞) in C for j ∈ N * . Following Lemma 11.1, it is easy to check that the functionsψ j are analytic on (0, +∞), positive, increasing, strictly convex and that: Letg j be the inverse ofψ j defined on (0, b j /c j ). In particular, for a given positive v < b j /c j , the minimum ofψ j (u) − uv for u ≥ 0 is uniquely reached atg j (v). Using thatψ j converges uniformly, on compacts sub-sets of a neighborhood in C of (0, +∞), towards ψ, thatb and thusψ j andψ are analytic, we get that for any compact of (0, +∞) and j large enough, the strictly convex functionsψ j and their derivatives converge uniformly towards the strictly convex functionψ and its derivatives. We deduce that for any compact K of (0, +∞) and j large enough (more precisely j such that b j /c j > sup(K)), g j is well defined on K and converges uniformly towardsg on K.
We consider the following general setting. Let ∈ N * and a n ∈ [ c n /c 0 , b n ) such that lim sup n→∞ a n / b n < 1. Since b > µ > c r+1 /c r for all r ∈ N, we deduce that the sequence (c n−l b l , 0 ≤ l ≤ n) is increasing. Therefore, the integer l n = sup{l ∈ {0, . . . , n}, c n−l b l ≤ c 0 a n } is well-defined and strictly less than n. Set j n = n − l n ≥ 1 and y n such that: a n = y n c jn b ln , (11.16) so that y n ∈ [1/c 0 , bc jn−1 /c 0 c jn ). Notice that the conditions lim n→∞ a n /c n = +∞ and a n < b n imply that lim n→∞ l n = +∞. The sequence (j n , n ∈ N * ) may be bounded or not.
As c r+1 /c r < b for all r ∈ N, we deduce that y n < bc jn−1 /c 0 c jn < b jn /c jn . Thus, we can defineũ * n, =g jn (y n ) andσ 2 n, =ψ jn (ũ * n, ) > 0. Lemma 11.5. Let p be a non-degenerate super-critical offspring distribution with b < ∞ and type (L 0 , r 0 ). Let ∈ N * . Assume that lim n→∞ a n /c n = ∞ and lim sup n→∞ a n / b n < 1. Then, we have, with lim n→∞εn, = 0: The proof, detailed in Section 11.8 is in the spirit of the proof of (175) in [18]. We end this section with the following strong ratio limit. Lemma 11.6. Let p be a non-degenerate super-critical offspring distribution with b < ∞ and type (L 0 , r 0 ). Let ∈ N * . Assume that lim n→∞ a n /c n = ∞, lim sup n→∞ a n / b n < 1, and a n = r n 0 (mod L 0 ) for all n ∈ N * . Then, we have: . (11.17) Proof. Let ∈ N * . Assume that a n ∈ [ c n /c 0 , b n ) and a n = r n 0 (mod L 0 ) for all n ∈ N * and lim sup n→∞ a n / b n < 1. An estimation of P (Z n = a n ) is given in Lemma 11.5. We now give an estimation of P (Z n = a n ) with n = n − h for some h ∈ N * and = b h . Recall (11.16) and the definition of l n , j n and y n . We have: a n = y n c j n b l n = y n c j n b l n +h , with j n + l n = n = n − h and l n = sup{l ∈ {0, . . . , n = n − h}, c n−h−l b l+h ≤ c 0 a n }. From the definition of l n , we deduce that l n = l n − h so that b l n = b ln , j n = j n and thus y n = y n . This gives thatg jn (y n ) =g j n (y n ) and thusũ * n , =ũ * n, as well asσ 2 n , =σ 2 n, .
Thanks to Remark 1.4, we have P b h (Z n−h = a n ) > 0 and P (Z n = a n ) > 0 for n large.

Second step: the main part
We now consider the main part J(t 0 ) = |t|≤t0 H(u + it) dt. An integration by part gives: (f r (φ(u + it)) − c) e −(u+it)yb r dt.
Using Lemma 11.2, we get there exists a constant C such that for all l, j ∈ N * and (u, t) ∈ A j : Using thatb is analytic and increasing on (1, +∞) and m 1 > 1, we deduce that there exists ε > 0 such that for all j ∈ N * , u ∈ K: We deduce that for all l, j ∈ N * , (u, t) ∈ A j : This gives that for all u ∈ K, l, j ∈ N * : We have H l,j (z) = φ j (z)f l (φ j (z)) f l (φ j (z)) −1 . For (u, t) ∈ A j , we have using (10.9), (11.29) and f l (|z|) ≤ b l f l (|z|)/|z|: Arguing as in the upper bound on I ± 1 , we get there exists a finite constant C such that for all l, j ∈ N * , u ∈ K Then use (10.12), to conclude that |I 2 | ≤ (C/y) e b l (ψ(u)−uy)− b l ε for some finite constant C. This and (11.30) give there exists a finite constant C such that for all l, j ∈ N * , u ∈ K: (11.31)

Conclusion
To conclude, we deduce from (11.31) with y = y n that: We present mostly the results without proof as their correspond either to a slight generalization of [17] and [18] or can be proven by mimicking the proof in the Harris case presented in Section 11. Recall the Böttcher constant β ∈ (0, 1) is defined by a = µ β , where a is the minimum of the support of p. We assume a ≥ 2.
Let P be the distribution of i=1 W i , with (W i , i ∈ N * ) independent random variables distributed as W . Since a > 0 and thus c = 0, we get that W has density w and that i=1 W i has density w * . Mimicking very closely the proof in [18] stated for = 1, it is not very difficult to check the following result. The verification is left to the reader. Lemma 12.1. Let p be a non-degenerate super-critical offspring distribution with finite mean and a ≥ 2. Let ∈ N * . As x 0, we have:   123) and (78) in [18], we also get the following upper bound, see also Lemma 11.3 in the Harris case.
Corollary 12.2. Let p be a non-degenerate super-critical offspring distribution with finite mean and a ≥ 2. There exists a finite constant C such that for all ∈ N * , x > 0 and u ≥ 0, we have with r = r(x), y = y(x): w * (x) ≤ Cµ r e uya r ϕ(u) f r (ϕ(u)) . (12.8)

Proof of Lemma 4.4 in the Böttcher case
Mimicking the arguments given in Section 11.3, it is easy, using Corollary 12.2 to get that: lim x→0+ µ w * a (x) w * (x/µ) p(a) = 1.

Lower large deviations for Z n
For j ∈ N * , let ϕ j denote the Laplace transform of W j = Z j /c j : ϕ j (u) = E[e −uWj ] = f j (e −u/cj ) for u ∈ C + , where C + = {u ∈ C, R(u) ≥ 0}. Notice that ϕ j converges uniformly on the compacts of C + towards ϕ, the Laplace transform of W , as j goes to infinity. We also have that ϕ j (u)/ϕ j (u) = −E[W j e −uWj ]/E[e −uWj ] so that lim u→+∞ ϕ j (u)/ϕ j (u) = −a j /c j .
In particular, for a given v > a j /c j , the minimum of ψ j (u) + uv for u ≥ 0 is uniquely reached at g j (v). Using that ψ j converges uniformly on compact of C + towards ψ, that b and thus ψ j and ψ are analytic, we get that for any compact of (0, +∞), the strictly convex functions ψ j and their derivatives converge uniformly towards the strictly convex EJP 24 (2019), paper 15. function ψ and its derivatives. We deduce that for any compact of (0, +∞), g j converges uniformly towards g. We consider the following general setting. Let ∈ N * and a n ∈ ( a n , c n /c 0 ] such that lim inf n→∞ a n / a n > 1. Since a < c r+1 /c r < µ for all r ∈ N, we deduce that the sequence (c n−l a l , 0 ≤ l ≤ n) is decreasing. Therefore, the integer l n = sup{l ∈ {0, . . . , n}, c n−l a l ≥ c 0 a n } is well-defined and strictly less than n. Set j n = n − l n ≥ 1 and a n = y n c jn a ln , with y n ∈ (ac jn−1 /c 0 c jn , 1/c 0 ]. Notice that the conditions lim n→∞ a n /c n = 0 and a n > a n imply that lim n→∞ l n = +∞. The sequence (j n , n ∈ N * ) may be bounded or not.
As a < c r+1 /c r for all r ∈ N, we deduce that y n > ac jn−1 /c 0 c jn > a jn /c jn . Thus, we can define u * n, = g jn (y n ) and σ 2 n, = ψ jn (u * n, ). Mimicking very closely the proof of (175) in [18] (which is stated for = 1 and lim n→∞ j n = ∞), it is not very difficult to check the following slightly more general result. The verification, which can also be seen as a direct adaptation of the detailed proof of Lemma 11.5, is left to the reader. Lemma 12.3. Let p be a non-degenerate super-critical offspring distribution with finite mean, a ≥ 2 and type (L 0 , r 0 ). Let ∈ N * . Assume that lim n→∞ a n /c n = 0 and lim inf n→∞ a n / a n > 1. Then, we have, with lim n→∞ ε n, = 0: P (Z n = a n ) = L 0 p(a) − /(a−1) c jn 2π a ln σ 2 n, exp a ln (ψ jn (u * n, ) + u * n, y n ) (1 + ε n, (1))1 {an= r n 0 (mod L0)} .
We end this section with the following strong ratio limit, whose proof is similar to the proof of Lemma 11.6. Lemma 12.4. Let p be a non-degenerate super-critical offspring distribution with finite mean and a ≥ 2. Assume that lim n→∞ a n /c n = 0, lim inf n→∞ a n / a n > 1 and a n = r n 0 (mod L 0 ) for all n ∈ N * . Then, we have: lim n→∞ P a h (Z n−h = a n ) P (Z n = a n ) = p(a) −(a h −1) /(a−1) . (12.9)

Proof of Proposition 6.5 in the Böttcher case
For h ∈ N, we have P(r h (τ ) = r h (t a )) = p(a) (a h −1)/(a−1) . We deduce from (2.6) and the convergence characterization (2.1), using that t a has a.s. an infinite height, that the proof of Proposition 6.5 is complete as soon as we prove the following strong ratio limit. Lemma 12.5. Let p be a non-degenerate super-critical offspring distribution with finite mean and such that a ≥ 2. Assume that lim n→+∞ a n /c n = 0 and that P(Z n = a n ) > 0 for every n ∈ N (which implies that a n ≥ a n ). Then, we have for h, k ∈ N * : lim n→∞ P k (Z n−h = a n ) P(Z n = a n ) = p(a) −(a h −1)/(a−1) 1 {k=a h } . (12.10) In fact, it is enough to prove (12.10) for k = a h as P(Z h = a h ) = p(a) −(a h −1)/(a−1) . It is also enough to consider the two cases: lim n→∞ a n /a n = 1 and lim inf n→∞ a n /a n > 1.
The case lim n→∞ a n /a n = 1 is handled as in the Harris case, see the first part of the proof of Proposition 6.3 in Section 6.2. The case lim inf n→∞ a n /a n > 1 is a consequence of Lemma 12.4. EJP 24 (2019), paper 15.