On the maximal offspring in a subcritical branching process

We consider a subcritical Galton--Watson tree conditioned on having $n$ vertices with outdegree in a fixed set $\Omega$. Under mild regularity assumptions we prove various limits related to the maximal offspring of a vertex as $n$ tends to infinity.


Introduction and main results
1.1. The largest degree. -Let T be a Galton-Watson tree whose offspring distribution ξ satisfies P(ξ = 0) > 0 and P(ξ ≤ 1) < 1. We assume that E[ξ] < 1 and that there is a slowly varying function f and a constant α > 1 such that P(ξ = n) = f (n)n −1−α (1.1) for all large enough integers n. For a fixed non-empty set Ω ⊂ N 0 of non-negative integers with P(ξ ∈ Ω) > 0 we may consider the tree T Ω n obtained by conditioning T on the event that the number of vertices with outdegree in Ω is equal to n. In order to avoid technicalities we assume that 0 ∈ Ω and that either Ω or its complement Ω c = N 0 \ Ω is finite. Setting θ = min(α, 2) we let (X t ) t≥0 be the spectrally positive Lévy process with Laplace exponent E[exp(−λX t )] = exp(tλ θ ). Let h be the density of X 1 . Our first main result is a local limit theorem for the maximal outdegree ∆(T Ω n ) of the tree T Ω n .
Theorem 1.1. -There is a slowly varying function g such that uniformly for all ∈ Z.
This contrasts the case of a critical Galton-Watson tree, where the largest outdegree has order o p (n) [8,9,22], although condensation may still occur on a smaller scale [27]. In the subcritical regime the special case T n := T N 0 n was studied by Janson [22,Thm. 19.34], who established a central limit theorem for ∆(T n ) if ξ follows asymptotically a power law. This was extended to   (1) The precise behaviour of the unique vertex with macroscopic degree in the center of the image is described by Theorems 1.1 and 1.2. The offspring distribution was chosen to be of the form (1.1) with α = 3/2, f (n) constant (except for the first term f (0)), and E[ξ] = 1/2. offspring laws with a regularly varying density by Kortchemski [26,Thm. 1] using a different approach, which also inspired the present work. Hence Theorem 1.1 generalizes this result to different kinds of conditionings and sharpens the form of convergence to a local limit theorem.
The case θ = 3/2 is related to extremal component sizes in uniform random planar structures. It was observed by Banderier, Flajolet, Schaeffer, and Soria [6,Thm. 7] that the Airy distribution precisely quantifies the sizes of cores in various models of random planar maps. This phenomenon was also observed in random graphs from planar-like classes by Giménez, Noy, and Rué [18,Thm. 5.4]. The local limit theorems established in these sources were obtained using analytic (1) Visualization was done with Mathematica using a spring-electrical embedding algorithm. The simulation of the random tree was carried out with the author's open source software GRANT -Generate RANdom Trees -available at: https://github.com/BenediktStufler/grant. The code implements a multithreaded version of Devroy's algorithm [13] for simulating size-constrained Galton-Watson trees.
methods. For uniform size-constrained planar maps and related models Addario-Berry [4,Thm. 3] observed that the number of corners in the 2-connected core is distributed like the largest outdegree in a simply generated tree. In a similar spirit S. [32, Cor. 6.42, Thm. 6.20] related the largest 2-connected block in random graphs from planar-like classes and general tree-like structures to a Gibbs partition of the maximum outdegree in large Galton-Watson trees. These connections have been used in [4,32] to recover the central limit theorem of the largest block in these models via a probabilistic approach. Theorem 1.1 enables us to strengthen these alternative proofs to recover the stronger local limit theorem.
1.2. The global shape. -We say a plane tree is marked if one of its vertices is distinguished. The path connecting the root to the marked vertex is called the spine. The fringe subtree of a plane tree at a vertex is the subtree consisting of the vertex and all its descendants. Any plane tree T may be fully described by the ordered list (F i (T )) 1≤i≤∆(T ) of fringe subtrees dangling from the lexicographically first vertex v with maximum outdegree, and the marked tree F 0 (T ) obtained from T by marking the vertex v and cutting away all its descendants.
We consider the size-biased random variableξ with values in N ∪ {∞} and distribution given by Let T • be the random marked plane tree constructed as follows. We start with a root vertex and sample an independent copyξ 1 ofξ. If it is equal to infinity, then we mark the root vertex and stop the construction. Otherwise we add offspring according toξ to the root vertex. We select one of the offspring vertices uniformly at random and declare it special. Each of the non-special offspring vertices gets identified with the root of an independent copy of the ξ-Galton-Watson tree T. We then iterate the construction with the special offspring vertex. In particular, the marked vertex of T • is always a leaf. For any finite plane tree T with a marked leaf v it holds that Let (T i ) i≥1 denote independent copies of the ξ-Galton-Watson tree T. The following observation describes the asymptotic shape of the conditioned Galton-Watson tree T Ω n . It tells us that F 0 (T n ) converges weakly to T • , and that all but a very small number of the fringe subtrees dangling from the vertex with maximum outdegree in T n behave asymptotically like independent copies of the unconditioned Galton-Watson tree T. Theorem 1.2. -There is a constant C > 0 such that for any sequence of integers (t n ) n≥1 with t n → ∞ it holds that Here d ≈ denotes that the total variation distance of the two random vectors tends to zero as n tends to infinity. We also let | · | denote the number of vertices. Results similar to Theorem 1.2 are known to hold for the tree T n = T N 0 n . Janson [22,Thm. 20.2] established convergence of (F 0 (T n ), F i (T n )) 1≤i≤k ) for k a constant, assuming significantly weaker requirements on ξ. Specifically, assuming only that ξ is heavy tailed and E[ξ] < 1 he showed that such a limit holds with respect to the lexicographically first vertex having "large" outdegree instead of maximum outdegree. The condition ensuring that "large" means maximum with high probability is ∆(T n ) = (1 − E[ξ] + o p (1))n, which seems to be more general than assumption (1.1). Kortchemski [26,Cor. 2.7] showed a limit for the fringe subtrees (F i (T n )) 1≤i≤(1−E[µ]− )n in the setting (1.1) for > 0 an arbitrarily small constant.
Abraham and Delmas [2] established a local weak limit for the vicinity of the root vertex of T Ω n in the more general condensation regime. This implies that a vertex with large outdegree emerges close to the root. In general this vertex does not have to correspond with high probability to a vertex with maximum outdegree of T Ω n . But it clearly does in the setting (1.1), yielding weak convergence of (F 0 (T Ω n ), F i (T Ω n )) 1≤i≤k ) for any fixed constant k. (More generally this holds if [22,Sec. 20]. In the setting (1.1) this is ensured by Theorem 1.1.) For conditioned Galton-Watson trees that encode certain types of Boltzmann planar maps a result concerning the asymptotic behaviour of the forest of fringe subtrees dangling from a vertex with macroscopic degree was also established by Janson and Stefánsson [23].
1.3. Limits of graph parameters. -We postpone the complex proofs of Theorems 1.1 and 1.2 to Sections 2 and 3. In the present Section we collect and prove limits of the height, the microscopic vertex degrees, and fringe subtrees. with H denoting the height of the marked vertex in T • . It follows a geometric distribution The family (T i ) i≥1 denotes independent copies of the unconditioned tree T, and M n denotes an independent random integer satisfying M n = n(1 − E[ξ])P(ξ ∈ Ω) −1 + O p (g(n)n 1/θ ). By extreme value statistics it follows that uniformly for all integers k ≥ 0 Similar as for the height in (1.5), it follows that the second largest degree ∆ 2 (T Ω n ) satisfies By extreme value statistics we obtain: We may additionally describe the asymptotic behaviour of ∆ i (T Ω n ) for each fixed i ≥ 2: In Section 3 we describe how T Ω n may be sampled by taking a simply generated treeT n with n vertices (with offspring distribution described in Equation (3.7)) and blowing up each vertex into an ancestry line by a process illustrated in Figure 6. This construction goes back to Ehrenborg and Méndez [15], and was fruitfully applied in the probabilistic literature [29,30,2,1]. Applying results from [5] and [26,Cor. 2.7]  Here (ξ i ) i≥1 denote independent copies of the branching mechanismξ with probability generating functionφ(z) stated in Equation (3.7). The blow-up procedure transformsξ into a depth-firstsearch order respecting segment of independent outdegrees. Here (ξ Ω c i ) i≥1 denotes independent copies of (ξ | ξ ∈ Ω c ), ξ Ω d = (ξ | ξ ∈ Ω), and L an independent geometrically distributed integer with distribution (1.14) (In case Ω = N 0 this means that L = 0 is almost surely constant.) The description of the blowup of the first coordinate n−1 i=1 (1 −ξ i ) is more delicate, and carried out in Lemma 3.1. We obtain: . . , d |T Ω n | denote the depth-first-search ordered list of vertex outdegrees of T Ω n , and let j 0 be the smallest index with d j 0 = ∆(T Ω n ). Let (D i ) i≥1 be independent copies of D. We let denote a binary operator that concatenates any two given lists.
This implies fine-grained information on the number |T Ω n | of vertices in T Ω n .
∈ Ω with high probability. Ordering n − 1 independent copies of ξ Ω in a descending manner ξ Ω (1) ≥ . . . ≥ ξ Ω (n−1) , it follows that ∈ Ω c with high probability. Letting (L i ) i≥1 denote independent copies of L, we may order n+1 i=1 L i many independent copies of ξ Ω c in a descending manner For the case Ω = N 0 and ξ following a power law, this was established by Janson [22,Thm. 19.34, (iii)].

1.3.3.
Counting fringe subtrees. -For any finite plane tree T we may consider the functional N T (·) that takes a plane tree as input and returns the number of occurrences of T at a fringe. It is easy to see that the unconditioned ξ-Galton-Watson tree T satisfies By Theorem 1.1 and 1.2 it follows that This agrees with known results for the case Ω = N 0 , see [22,Thm. 7.12].
We may also derive exponential concentration inequalities: Let F (·) denote a function that takes a list of at least |T | integers as input, extends it cyclically, and returns the number of occurrences of the depth-first-search ordered list of outdegrees of T as a substring of the cyclically extended input. Note that such substrings cannot overlap. Hence changing a single coordinate of the input changes the value of F by at most 1.
Recalling that d 1 , . . . , d |T Ω n | denotes the outdegree list corresponding to T Ω n , we may write Letting (ξ) i≥1 denote independent copies of ξ, it follows by McDiarmid's inequality that for all x > 0 and ≥ |T | Here we have used that E[F (ξ 1 , . . . , ξ )] = P(T = T ), since F cyclically extends the input list. The number L Ω (T) of vertices of the unconditioned ξ-Galton-Watson tree T with outdegree in Ω is known to equal the total of number of vertices in a Galton-Watson tree with a different offspring distribution [30] (that is critical/subcritical heavy-tailed/light-tailed if and only if ξ is). See Section 3 for details. Hence it follows by a general result of Janson [22] that We may write As plane trees correspond to cyclic shifts of balls-in-boxes configurations (see Equation (2.1)), the Chernoff bounds and Equation (1.26) imply that this bound simplifies to exp(−Θ(n)). Using Equations (1.24) this implies By Equation (1.25) this bound simplifies to exp(−Θ(n)). We obtain: This agrees with the known case Ω = N 0 , see for example [32, Thm. 6.5].
Remark 1.9. -The proof of Lemma 1.8 does not use any of the assumptions on ξ. Hence Inequality (1.27) holds for any offspring distributions ξ (with P(ξ = 0) > 0, P(ξ ≥ 2) > 0, and P(ξ ∈ Ω) > 0), that is either critical, or subcritical and heavy-tailed. Furthermore, it is well-known that if ξ subcritical and light-tailed, then the study of T Ω n may be reduced to one of these two cases by tilting the offspring distribution.

Lemma 1.8 implies by the Borel-Cantelli Lemma that
This may be expressed equivalently in terms of random probability measures. Take the random tree T Ω n , and let µ n denote the (random) law of the fringe subtree at a uniformly selected vertex of T Ω n . Then Similar as for the height in (1.5), we obtain by extreme value statistics the following limit for the maximal size of the fringe subtrees F i (T Ω n ), 1 ≤ i ≤ ∆(T Ω n ) dangling from the vertex with maximum degree.
in the space D([0, 1], R) for some slowly varying function f 1 (n). Hence: This was established by Kortchemski [26,Thm. 3] for the case Ω = N 0 .  The tree in Figure 2 exhibits a unique vertex whose degree has order (1 − E[ξ])n, which is typical for the regime [24,22,26,1] of Theorem 1.1. A similar condensation phenomenon has recently been shown to occur when ξ is critical and lies in the domain of attraction of Cauchy law [27], see Figure 3 for an illustration. There the order of the maximum degree is o(n), but varies regularly with index 1, and is much larger than the second largest degree. In the so called stable regime illustrated in Figure 4, ξ is critical and lies in the domain of attraction of an α-stable law for 1 < α < 2. There for each fixed i ≥ 1 the order of the ith largest degree varies regularly with index 1/α [22,14,11]. Finally, the regime where ξ is critical and lies in the domain of attraction of the normal law [22,Sec. 19] is illustrated in Figure 5.

Conditioning on the number of vertices
We start by establishing the limit theorems for the special case Ω = N 0 using results by Denisov, Dieker, and Shneer [12] on the big-jump domain for random walks.
2.1. Plane trees correspond to cyclic shifts of balls-in-boxes configurations. -A (planted) plane tree T is a rooted unlabelled tree where each offspring set is endowed with a linear order. The outdegree of a vertex v ∈ T , denoted by d + T (v), is its number of children. We let ∆(T ) denote the maximal outdegree of T . The total number of vertices of T is denoted by |T |. Setting n = |T |, the tree T is fully determined by the vector with v 1 , . . . , v n denoting the depth-first-search ordered list of vertices of T . The vector (x i ) 1≤i≤n satisfies n i=1 x i = −1 and k i=1 x i ≥ 0 for all k < n. The following result is classical: satisfies k i=1ȳ i > −r for all k < n.
Hence for r = 1 such a vector y corresponds to a unique tree T (y). The index i 0 is obtained by letting k 0 denote the smallest integer between 1 and n for which and setting i 0 = 1 if k 0 = n, and i 0 = k 0 + 1 otherwise.

2.2.
Non-generic simply generated trees and the big-jump domain. -If (ξ i ) i≥1 denote independent copies of ξ, then Let v 1 , . . . , v n denote the depth-first-search ordered list of vertices of T n , and set d i = d + Tn (v i )− 1. (The depth-first-search order is often also referred to as the lexicographic order due to the usual embedding of plane trees as subtrees of the Ulam-Harris tree.) Let 1 ≤ j 0 ≤ n denote the smallest index such that the maximum outdegree of T n is attained at the corresponding vertex. It was observed in [26, Cor. 2.7] using results from [5] (compare with [22,Thm. 19.34, (iii)]) that Note that v n does not have to correspond to a tree, since the first coordinate may be smaller than −1. In this case, we set T (v n ) = for some symbol that is not contained in any other set under consideration in this paper. The probability for this event tends to zero as n becomes large. Equation This may be used (see [26,Thm. 1]) to deduce a central limit theorem Compare also with [22,Thm. 19.34]. We may strengthen (2.7) to a local limit theorem. This does not follow directly from (2.6), as we would require knowledge on the speed with which the total variation distance tends to zero.
uniformly for all integers .
It follows from Equations (2.8), (2.9) and the exponential bounds [12, Lem. 2.1] (applied to the centred random walk S n − nE[ξ]) that there is a constant C > 0 such that for all c ≥ 1. Hence there is a constant 1 > 0 such that it suffices to verify that Lemma 2.2 holds uniformly for all ≥ 1 n/ log n.
Throughout the following we only consider values with 1 n/ log n ≤ ≤ n. By Equations (2.8) and (2.9) it follows that g(n)n 1/θ P(∆(T n ) = ) equals Our next step is to discard all summands except for the first. Note that S n−k ≥ 0 implies that all summands with k > n/ are equal to zero. Hence Note that n/ ≤ −1 1 log n and hence n ∼ n − k uniformly for all summands. Since h is bounded, it follows from the local limit theorem (2.5) that P(S n−k = n − 1 − k )g(n)n 1/θ remains bounded uniformly for all 2 ≤ k ≤ n/ and ≥ 1 n/ log n. Moreover, nP(ξ = ) = O(n −α (log n) 1+α ) holds uniformly as well. Hence the expression in (2.11) may be bounded by This verifies that g(n)n 1/θ P(∆(T n ) = ) equals uniformly for all with 1 n/ log n ≤ ≤ n.
Let 0 < < 1 − E[ξ] be some constant. By [12, Cor. 2.1] (applied to the centred sum S n − nE[ξ]) it holds uniformly for all integers with 1 n/ log n ≤ ≤ that Since α > 1 implies that 1/θ < α, it follows that the expression in (2.12) tends to zero uniformly for all in the restricted range. Thus it suffices to verify that Lemma 2.2 holds uniformly for all with n ≤ ≤ n. Let ∈ [ n, n] be given. Our next step will be to get rid of the event max 1≤i≤n−1 ξ i < in the expression (2.12). To this end, note that sup k≥n P(ξ = k) = O(P(ξ = n)) implies that Also, remains bounded for ≥ n. Using again 1/θ < α, this implies that the result of substituting max 1≤i≤n−1 ξ i < by max 1≤i≤n−1 ξ i ≥ in expression (2.12) tends to zero. This shows that g(n)n 1/θ P(∆(T n ) = ) = o(1) 1+α P (S n−1 = n − 1 − ) g(n)n 1/θ (2.14) holds uniformly.
The local limit theorem (2.5) tells us that Using that the function h and the expression in (2.13) are bounded, it follows from (2.14) that Let 2 > 0 be small enough such that 1/θ + 2 < 1. It holds that Consequently, it remains to verify that Lemma 2. This completes the proof.
2.4. The asymptotic shape of the random tree T n . - Proof. -We may assume that t n ≤ log n for all n. As ∆(T (v)) = (1 − E[ξ] + o p (1))n this ensures that t n < ∆(T n ) with high probability. Since F i (T n ) = n and F i (T n ) ≥ 1 for all i it follows that in this case We are going to show that holds uniformly for all marked plane trees T • ∈ T • and all ordered forests (T i ) 1≤i≤k of plane trees with total number of vertices |T Recall that Section 2.1 we discussed how plane trees correspond to sequences (x i ) 1≤i≤m with m i=1 x i = −1 and i=1 x i ≥ 0 for all < n. An ordered forest of plane trees corresponds to concatenations of such sequences. There is a unique way to cut (ξ 1 − 1, . . . , ξ n−1 − 1) into initial segments x 1 , . . . , x r , each corresponding to a tree, and a single tail segment y = (y i ) 1≤i≤d with j i=1 y i ≥ 0 for all 1 ≤ j ≤ d. (For example, x 1 corresponds to the tree F 1 (T (v)).) The segment y corresponds to the initial segment of the depth-first-search ordered list of vertex outdegrees of F 0 (T (v n )) obtained by stopping right before visiting the lexicographically first vertex with maximum outdegree. Hence it encodes the outdegrees of the spine vertices (except for the marked vertex), and all vertices that lie to the left of the spine. It also encodes the precise location of the marked vertex. The sum R := d i=1 y i tells us the quantity of direct offspring of spine vertices (except for the marked vertex) of F 0 (T (v n )) that lie to the right of the spine. Hence x 1 , . . . , x r−R correspond to the fringe-subtrees dangling from the marked vertex in T (v n ), and x r−R+1 , . . . , x r correspond to the fringe subtrees dangling from spine vertices (except for the marked vertex) to the right of the spine. So in order for the tail segment x r−R+1 , . . . , x r , y to encode the tree T • it must holds that the concatenation of y, −1, x r , x r−1 , . . . , x r−R+1 is equal to the depth-first-search ordered list of outdegrees of T • . In order for x 1 , . . . , x k to encode (T i ) 1≤i≤k it must hold that x i encodes T i for all 1 ≤ i ≤ k. The only requirement for the middle segment (ξ 1+ k i=1 |T i | , . . . , ξ n−|T • | ) is that it must correspond to a forest. Note that the middle segment has S n := n − |T • | − k i=1 |T i | list entries and S n ≥ t n by assumption. Using Equation (1.3) it follows that the probability 1 − 1, . . . , ξ Sn − 1) corresponds to a forest).

By Equations (2.4) and (2.16) this implies that (F
By splitting the event that (ξ 1 − 1, . . . , ξ Sn − 1) corresponds to a forest according to the number of trees in the forest we obtain that its probability is given by By Lemma 2.1 we may simplify this expression to The sum 1≤i≤Sn (ξ i − 1) is a random walk with negative drift.
Since S n ≥ t n and t n → ∞ it follows that uniformly in T • and (T i ) 1≤i≤k as n tends to infinity. Hence holds uniformly. This verifies (2.17) and hence completes the proof.

Conditioning on the number of vertices with outdegree in the set Ω
We reduce the case of a general Ω to the special case Ω = N 0 via a combinatorial transformation. This construction goes back to Ehrenborg and Méndez [15] and is also known in the probabilistic literature, see Abraham and Delmas [2,1], Minami [29], and Rizzollo [30]. Further studies of related conditionings of Galton-Watson trees may be found in [25,3,10].
Throughout this chapter we assume that Ω is a proper, non-empty subset of N 0 such that either Ω or its complement Ω c := N 0 \ Ω is finite. To each finite (planted) plane tree T we may assign its weight ω(T ) = P(T = T ). We let L Ω (T ) denote the number of vertices in T whose outdegree lies in Ω. The generating function with the index T ranging over all finite plane trees represents the combinatorial class A of plane trees weighted by ω and indexed according to the number of vertices with outdegree in Ω. (2) We set ω k = P(ξ = k) and for any subset M ⊂ N 0 we set Decomposing with respect to the outdegree of the root vertex readily yields Since 0 lies in Ω then we may write We may interpret this equation as follows. If the root vertex has outdegree in Ω, then we have to account for it by a factor z and attach the roots of a weighted forest ϕ(A(z)). This accounts for the first summand. The second corresponds to the case where the outdegree of the root does not lie in Ω. Here we take a root-vertex, attach to it as left-most offspring the root of a tree (counted by A(z)) and then add the root of a weighted forest φ * Ω c (A(z)) as siblings to the right. If we are in the second case, then we may recurse this case-distinction at the left-most offspring of the root. In this way, we descend along the left-most offspring until we encounter for the first time a vertex with outdegree in Ω. In this way we form an ordered list of φ * Ω c (A(z))-forests, yielding .

(3.7)
In combinatorial language, decomposition (3.7) identifies the class A as the class ofφ-enriched plane trees. We refer the reader to [32] and references given therein for details on the enriched trees viewpoint on random discrete structures.
We letξ denote a random non-negative integer with distribution given by the probability generating seriesφ. We letT denote aξ-Galton-Watson tree and letT n = (ξ | |ξ| = n) denote the result of conditioning it to have n vertices. For each k ≥ 0 let A k denote the set of all vectors (2) A (weighted) combinatorial class consists of a countable set S and a weight function γ : S → R ≥0 . The class may be indexed by a size function s : S → N0 and the corresponding generating series may be formed by (y, x 1 , . . . , x ) with ≥ 0, y ∈ Ω, x 1 , . . . , x ∈ Ω c − 1, and y + i=1 x i = k. We let the weight of such a vector be given by For each vertex v ∈T n we independently select a vector β n (v) ∈ A d +

Tn
(v) at random with probability proportional to its weight. The pair (T n , β n ) is aφ-enriched plane tree with n vertices that by the decomposition (3.7) corresponds to a plane tree that has precisely n vertices with outdegree in the set Ω. The correspondence goes by blowing up each vertex v ∈T n into an ancestry line according to β n (v) as illustrated in Figure 6. The blow-up of the random enriched plane tree (T n , β n ) is distributed like the random tree T Ω n . This may be verified directly as by Rizzolo [30] or deduced from a general sampling principle [32, Lem. 6.1]. Note that E[ξ] < 1 implies that .
. . , X k L k ) be drawn from A k with probability proportional to the weights defined in (3.8). We form the sequence (Y k , X k 1 , . . . , * , . . . , X k L k ) by replacing the largest coefficient in the sequence (Y k , X k 1 , . . . , X k L k ) with a * -placeholder. Let L, X, and Y be random independent integers with distributions given by , Let (X i ) i≥1 and (X i ) i≥1 be independent copies of X, and let L be an independent copy of L.

a)
If Ω c is finite, then as k tends to infinity. b) If Ω is finite, then as k tends to infinity. c) There are constantsk, C, c > 0 such that for all k ≥k and x ≥ 0 it holds that . (3.14) Proof. -Claim a) is a probabilistic version of the enumerative result (3.11) and follows by standard arguments. Claim b) is the probabilistic version of the enumerative formula (3.11) and may be justified using a general result for the asymptotic behaviour of random Gibbs partitions that exhibit a giant component [31,Thm. 3.1].
Claim c): Suppose that |Ω| < ∞. In this case it holds by [17,Thm. 4.30] that as k becomes large. It follows that there are constants C 2 , k 0 > 0, such that for all k ≥ k 0 and x ≥ 0 Applying the bound [17,Thm. 4.11] yields that for any > 0 there are constants c( ), 0 > 0 such that for all ≥ 0 Using that L has finite exponential moments it follows that there are constants c 1 , C 3 > 0 such that P(max(X 1 , . . . , X L ) = k | X 1 + . . .
Since we are in the case |Ω| < ∞ the random variable Y has a deterministic upper bound, and it follows that there are positive constants C 4 , k 1 > 0 with for all k ≥ k 1 and x ≥ 0.
In the case |Ω c | < ∞ the X i are deterministically bounded and the sum L + X 1 + . . . + X L has finite exponential moments. Hence, as k → ∞ This implies that there are constants k 2 , C 5 > 0 such that for all k ≥ k 2 and x ≥ 0 Using that X 1 + . . . + X L has finite exponential moments it follows that there are constants C 6 , c 2 > 0 such that as n tends to infinity. The limit (3.17) tells us that the first ∆(T n ) − t n fringe subtrees dangling fromṽ * become independent copies of theξ-Galton-Watson treeT. Given ∆(T n ) the family of fringe subtrees (F i (T n )) 1≤i≤∆(Tn) is conditionally exchangeable. Hence reordering the fringe subtrees in a suitable way and applying (3.17) yields that simultaneously the fringe subtrees dangling from the vertices belonging to small components in β n (ṽ * ) and the first Y ∆(Tn) − t n fringe subtrees corresponding to the large component become independent copies ofT. See the right hand of Figure 7 for an illustration.
The limit (3.17) also tells us that the total number of vertices of the remaining t n fringe subtrees inT n is with high probability smaller than 2t n /(1 − E[ξ]). When blowing up a tree we add additional vertices, but the size of any fringe subtree may at most double. This shows that the size of the blow ups of the remaining t n fringe subtrees is with high probability smaller than 4t n /(1 − E[ξ]).
The limit of F 0 (T Ω n ) is determined byT • together with the small components of β n (ṽ * ) and their fringe subtrees. Let us make this precise. Note that the blow-up ofT with canonically chosen random local decorations is by construction distributed like the ξ-Galton-Watson tree T. Let S • denote the random marked tree constructed as follows. We start with the blow-up of the treeT • with canonically chosen random decorations. If L = 0 we stop. Otherwise we add X L + 1 offspring vertices to the marked leaf. All except the first of these offspring vertices become the roots of independent copies of T. If L = 1 we declare the first offspring vertex to be the new marked leaf and stop the construction. Otherwise the first offspring vertex becomes father of X L−1 + 1 children. All but the first become roots of independent copies of T, and we proceed in the same manner until we are finished after L steps in total. Summing up, we have shown so far that The distribution of the random marked plane tree S • agrees with the distribution of T • . This follows by a slightly tedious but inoffensive calculation from standard properties of size-biased geometric distributions. It remains to treat the case |Ω| < ∞. This is analogous to the case |Ω c | < ∞, with the only difference being that we additionally have to take into account the small decorations X 1 , . . . , X L . That is, we have to check that the circled fringe subtree on the left hand side of Figure 7 follows the distribution of the Galton-Watson tree T. But this is clear, since it is distributed like the blow up ofT and hence like T.
Proof of Theorem 1.1. -Equations (3.11) and (3.9) allow us to apply Lemma 2.2 to the treẽ T n , yielding that there is a slowly varying function g with uniformly in x ∈ Z. We are going to show that as n tends to infinity. As h(t) → 0 as |t| → ∞, t ∈ R this already implies that (3.19) also holds uniformly for ∈ Z.
Our first step in the verification (3.19) is a lower bound on the maximum degree ∆(T Ω n ). By Equation (2.10) it follows that g(n)n 1/θ P(∆(T n ) ≤ n/ log 2 n) = o(1). (3.20) If we let Z n denote the size of the largest outdegree produced by blowing up the lexicographically first vertexṽ with maximal outdegree inT n , then it follows by (3.20), (3.18) and the fact that h is bounded that g(n)n 1/θ P(Z n ≤ n/ log 4 n) = o(1) + g(n)n 1/θ n/ log 2 n≤r≤n P(∆(T n ) = r, Z n ≤ n/ log 3 n) n/ log 2 n≤r≤n P(max(Y r , X r 1 + 1, . . . , X r Lr + 1) ≤ n/ log 4 n).
It follows easily by Equation (3.14) that this bound tends to zero. This shows that g(n)n 1/θ P(∆(T ω n ) ≤ n/ log 4 n) = o(1). (3.21) Thus, it suffices to verify that (3.19) holds with ranging over the set I n of integers in the interval from n/ log 4 n to n instead. To this end, let t n be a sequence that tends to infinity and let ∈ I n be an integer. Then g(n)n 1/θ P(∆(T Ω n ) = ) = R n, + S n, (3.22) with an error term R n, and S n, = 0≤x≤tn g(n)n 1/θ P(∆(T n ) = + x)P(max(Y +x , X +x 1 + 1, . . . , X +x L +x + 1) = ) the product of g(n)n 1/θ with the probability for the event that the largest outdegree in the blow-up of the lexicographically first vertexṽ with maximal outdegree inT n is equal to and that ∆(T n ) − ≤ t n . If this event fails but ∆(T Ω n ) = , then at least one of the following events must take place.
1) The maximal outdegree in the blow-up of the vertexṽ equals but ∆(T n ) − > t n . We let R n, (1) denote the product of g(n)n 1/θ with the probability for this event. 2) At least two vertices ofT n have outdegree at least and the blow-up of one of them produces a vertex with outdegree equal to . The product of g(n)n 1/θ with the probability for this event is denoted by R n, (2).
Hence R n, ≤ R n, (1) + R n, (2). (3.23) We are going to verify that this bound tends to zero uniformly for all ∈ I n . Using Inequality (3.14) and Equality (3.18) it follows that R n, (1) ≤ tn≤x≤n g(n)n 1/θ P(∆(T n ) = + x)P(max(Y +x , X +x 1 + 1, . . . , X +x L +x + 1) = ) and thus R n, (1) → 0 (3.24) uniformly for all ∈ I n . Here we have used that h is bounded, that the o(1) term tends to zero uniformly, and that the representation theorem for slowly varying functions implies that for any > 0 there is a positive constant C( ) with f ( ) f ( (1 + x )) ≤ C( ) 1 + x for all , x ≥ 1.
Using the local limit theorem (2.5) (forS n instead of S n ) and the fact that the density h is bounded it follows that We now turn our attention to S n, . By the limits (3.12) and (3.13) it follows that there is a probability density (p x ) x≥0 such that for each constant integer x ≥ 0 it holds that lim k→∞ P(max(Y k+x , X k+x 1 + 1, . . . , X k+x L k+x + 1) = k) = p x .