Graph limits of random graphs from a subset of connected $k$-trees

For any set $\Omega$ of non-negative integers such that $\{0,1\}\subseteq \Omega$ and $\{0,1\}\ne \Omega$, we consider a random $\Omega$-$k$-tree ${\sf G}_{n,k}$ that is uniformly selected from all connected $k$-trees of $(n+k)$ vertices where the number of $(k+1)$-cliques that contain any fixed $k$-clique belongs to $\Omega$. We prove that ${\sf G}_{n,k}$, scaled by $(kH_{k}\sigma_{\Omega})/(2\sqrt{n})$ where $H_{k}$ is the $k$-th Harmonic number and $\sigma_{\Omega}>0$, converges to the Continuum Random Tree $\mathcal{T}_{{\sf e}}$. Furthermore, we prove the local convergence of the rooted random $\Omega$-$k$-tree ${\sf G}_{n,k}^{\circ}$ to an infinite but locally finite random $\Omega$-$k$-tree ${\sf G}_{\infty,k}$.

However, other interesting graph classes like planar graphs or bipartite graphs are not partial k-trees. On the other hand, partial k-trees are very interesting from a combinatorial point of view, although the enumeration of partial k-trees for general k is still missing. The k-trees are the maximal graphs with treewidth ≤ k, in the sense that adding another edge would increase the treewidth. The number of k-trees has been counted in various ways; see [11,18,29,30,32,34,40,41,56]. As usual a graph on n vertices is called labelled if the integers from {1, 2, … , n} are assigned to its vertices (one-to-one).
In this paper, we introduce a subset of connected labelled k-trees, called Ω-k-trees, as a first attempt to approach the profile of connected labelled partial k-trees by using the enumeration of labelled k-trees. In what follows, without specifying otherwise, we assume that Ω-k-trees are all labelled and a random Ω-k-tree is uniformly selected from the class of labelled Ω-k-trees with (n + k) vertices.
Definition 1 (Ω-k-tree). For any set Ω of non-negative integers which contains 0,1 and at least one integer greater than 1, an Ω-k-tree is a connected k-tree such that the number of (k + 1)-cliques that contain any fixed k-clique belongs to the set Ω.
A rooted Ω-k-tree is an Ω-k-tree rooted at a k-clique. If Ω = N 0 = {0, 1, 2, …}, an N 0 -k-tree is a k-tree. See Figure 1 for an example of an Ω-2-tree. We remark that it is necessary to allow 0 ∈ Ω since by construction a k-clique is the smallest k-tree. We also need the condition 1 ∈ Ω. Otherwise any k-tree, other than a single k-clique, is infinite. We disregard the case Ω = {0, 1} in order to consider just non-trivial classes.
Darrasse and Soria [18] proved that the expected distance between pairs of vertices in a random k-tree with n vertices converges to a Rayleigh distribution after rescaling it by 1∕ √ n. The Rayleigh distribution also arises as the distance between typical vertices in Aldous' continuum random tree (CRT) [3][4][5]. This motivates the question whether the classical convergence of 1-trees to the CRT may be extended to the case k ≥ 2 and to models of random partial k-trees.
Since Aldous's pioneering work on the Galton-Watson trees, the CRT has been established as the limiting object of a large variety of combinatorial structures [12,14,16,17,39,44,54,57,58,61]. A key idea in the study of these combinatorial objects is to relate them to trees endowed with additional structures by using an appropriate bijection. In the present case of Ω-k-trees, we encode them as so-called (k, Ω)-coding trees via a bijection due to Darrasse and Soria [18], which was originally used to enumerate k-trees and to recursively count the distance between any two vertices in a random k-tree. Furthermore, in order to build a connection between the distance of two vertices in a random Ω-k-tree and the distance of two vertices in a critical Galton-Watson tree, we introduce the concept of a size-biased ( • , • )-multitype Galton-Watson tree. This is adapted from the size-biased Galton-Watson tree which was defined by Kesten [47], used by Lyons and coworkers in [52], by Addario-Berry, Devroye and Janson in [1], and was further generalized to the size-biased -enriched tree by Panagiotou and coworkers [58].
When we analyze Ω-k-trees, it turns out that it is convenient to consider the number of hedra instead of the number of vertices as the size of an Ω-k-tree (we adopt the notions from [34]). A hedron is a (k + 1)-clique in an Ω-k-tree, and by definition an Ω-k-tree with n hedra has (n + k) vertices. A front of a k-tree is a k-clique.
Our first main result establishes the convergence of a random k-tree to the CRT with respect to the Gromov-Hausdorff-Prokhorov distance.
Theorem 1 Let  n,k be the class of labelled Ω-k-trees with n hedra and denote by G n,k a random Ω-k-tree that is uniformly selected from the class  n,k . Let G n,k be the uniform measure on the set of vertices of  n,k . Then holds with respect to the Gromov-Hausdorff-Prokhorov metric. Here dist G n,k is the graph distance of G n,k , H k = 1 + 1∕2 + … + 1∕k denotes the kth harmonic number and Ω is a positive constant. If Ω = N 0 , then the constant N 0 equals 1.
This theorem shows in particular that the diameter and the distance of two independently selected random vertices in a random Ω-k-tree G n,k are of order √ n, and that up to scaling factors we obtain the same limiting distribution as that for random 1-trees. In fact, by general properties of the Gromov-Hausdorff-Prokhorov metric [55,Prop. 10], this statement may be extended to the case of any constant number of independent and uniform random vertices. Our proofs also show that the diameter of the random k-tree G n,k may be stochastically bounded by the diameter of a critical Galton-Watson tree conditioned to be large, where the offspring distribution has finite exponential moments. Hence the corresponding tail-bounds for the height of large Galton-Watson trees [1] allow us to deduce arbitrarily large uniform integrability for the rescaled diameter and rescaled distance of two random vertices in G n,k . Together with the distributional limits, this yields precise asymptotic expressions for all moments.
Instead of the class  n,k , we could equivalently consider the class of Ω-k-trees with n hedra that are rooted at a fixed labelled front. In section 2.1 we will argue that the two models are equivalent and hence our results apply to both.
We recall that (partial) 1-trees are just trees and partial 2-trees are series-parallel graphs. In both cases it is known [5,57] that the CRT appears as the scaling limit (if we scale the metric by c∕ √ n for some positive constant c). We conjecture that the CRT also arises as the scaling limit of partial k-trees for larger k. At the moment, this property seems to be out of reach, since there is no precise asymptotic analysis of partial k-trees if k ≥ 3.

Conjecture 1.
Let  n,k be the class of all connected labelled partial k-trees and let PT n,k be a uniform random graph from  n,k . Then, for every k ≥ 1, PT n,k converges toward the CRT in the Gromov-Hausdorff-Prokhorov sense, after rescaling the metric by a factor c k ∕ √ n for some constant c k > 0. Theorem 1 describes the asymptotic global metric properties of random k-trees, but gives little information about asymptotic local properties. Hence we provide a second limit theorem that establishes local weak convergence of the random Ω-k-tree G n,k toward an infinite but locally finite Ω-k-tree G ∞,k . This type of convergence describes the asymptotic behaviour of neighborhoods around a uniform random front.
n,k be the random front-rooted Ω-k-tree that is obtained by marking a uniform random front of the random Ω-k-tree G n,k . Then, as n tends to infinity, the random graph G • n,k converges in the local-weak sense toward a front-rooted infinite Ω-k-tree G ∞,k , that is, Our proof of Theorem 2 builds on the classical local convergence of simply generated trees toward a modified Galton-Watson tree. See for example Theorem 7.1 in Janson's survey [42], which unifies some results by Kennedy [46], Aldous and Pitman [6], Grimmett [36], Kolchin [49], Kesten [47], Aldous [4], Jonsson and Stefánsson [45] and Janson, Jonsson and Stefánsson [43].
A result similar to Theorem 2 is known for neighborhoods of random vertices in partial 2-trees since series-parallel graphs belong to the family of subcritical graph classes [33,60,62]. This motivates the following conjecture.

Conjecture 2.
For every k ≥ 1, the random labelled partial k-tree PT n,k converges in the local-weak sense. That is, the neighborhoods of a random vertex in PT n,k converge weakly toward the neighborhoods of an infinite rooted partial k-tree PT ∞,k as n tends to infinity.
The plan of the paper is as follows. In Section 2 we recall the combinatorial background for Ω-k-trees, introduce a multitype Galton-Watson process in order to uniformly generate a random (k, Ω)-coding tree, describe Darrasse and Soria's algorithm to compute the distances between two vertices in an Ω-k-tree, present Aldous's result on the convergence of critical Galton-Watson trees to the CRT  e , and recall the notion of local convergence. In Section 3 we prove our first main result-Theorem 1, and in Section 4 our second main result-Theorem 2.

COMBINATORICS, MULTITYPE GALTON-WATSON TREES AND GRAPH LIMITS
For any integer i ≥ 0, we set [i] ∶= {1, 2, … , i}. Let Ω ⊂ N 0 denote a set of non-negative integers which contains 0,1 and at least one integer greater than 1. We will review the generating function approach from [18] to count the number of Ω-k-trees. The key ingredient is a bijection between rooted Ω-k-trees and (k, Ω)-coding trees. Definition 2 ((k, Ω)-coding tree). For any set Ω of non-negative integers which contains 0,1 and at least one integer greater than 1, a (k, Ω)-coding tree of size n is a tree T consisting of kn + 1 white nodes and n black nodes which satisfies the following conditions: 1. T is rooted at a white node. Every white node has an unordered list of black nodes as children and every black node has an ordered list of precisely k white nodes as children. 2. The number of black children of the white root belongs to the set Ω and the number of black children of any other white one lies in the shifted set 3. The white root of T is labelled by a strictly increasing sequence (a 1 , The black nodes are labelled by the integers from the set [n + k] ⧵ A. 4. We label non-root white nodes of T recursively: if a black node is labelled with r and it is a child of a white node labelled with the sequence (r 1 , r 2 , … , r k ), then starting from the left, the ith child of the black node r is labelled with the sequence (r 1 , … , r i−1 , r, r i+1 , … , r k ), which is a sequence obtained from (r 1 , r 2 , … , r k ) by replacing r i by r.
In this way the labels on the white root and on the black nodes determine the labels on the non-root white nodes. If the white root of a (k, Ω)-coding tree C has precisely one black child, we call C a reduced (k, Ω)-coding tree.
The following classes and random graphs will play a key role in our arguments: 1.  n,k : the class of labelled Ω-k-trees with n hedra. 2. G n,k : a random Ω-k-tree that is uniformly selected from the class  n,k . 3.  • n,k : the class of labelled Ω-k-trees with n hedra that are rooted at a front. 4. G • n,k : a random Ω-k-tree G n,k that is rooted at a uniformly chosen front. This is equivalent to uniformly selecting an element from  • n,k . 5.  □ n,k : the class of labelled Ω-k-trees with n hedra that are rooted at a fixed front [k]. 6.  • n,k : the class of labelled Ω-k-trees with n hedra that are rooted at a fixed front [k] and this root front is contained in only one hedron. 7.  n,k : the class of (k, Ω)-coding trees with n black vertices, such that the white root is labelled with (1, 2, … , k). 8. C n,k : a random (k, Ω)-coding tree that is uniformly selected from  n,k . 9.  n,k : the class of reduced (k, Ω)-coding trees with n black vertices, such that the white root is labelled with (1, 2, … , k). 10. B n,k : a random reduced (k, Ω)-coding tree that is uniformly selected from  n,k . 11. G • n,k : a random Ω-k-tree that uniquely corresponds to B n,k under the bijection from section 2.2. This is equivalent to choosing a random Ω-k-tree uniformly from the class  • n,k . 12. G □ n,k : a random Ω-k-tree that uniquely corresponds to C n,k under the bijection . This is equivalent to choosing a random Ω-k-tree uniformly from the class  □ n,k .

Comparison of different rooting procedures
Any Ω-k-tree with n hedra has precisely kn + 1 fronts. Hence it makes no difference whether we uniformly select an element from the class  n,k or from the class  • n,k of labelled Ω-k-trees with n hedra that are rooted at a front. Instead of studying the random graph G n,k (as in Theorem 1) or the random graph G • n,k (as in Theorem 2), it suffices to consider uniformly selected elements from the class  • n,k . We can even make another simplification: For any k-subset A = {a 1 , a 2 , … , a k } ⊂ [n + k] we may consider the subset M A ⊂  • n,k of elements where the vertices of the root-front are labelled with a 1 , a 2 , … , a k . Given any other k-subset Thus,  • n,k is the disjoint union of ( n+k k ) -many relabelled versions of the class  □ n,k ∶= M [k] , where the root-front is required to be labelled from 1 to k. Hence, instead of studying G n,k or G • n,k , it suffices to study uniform elements from the class  □ n,k .

Correspondence between k-trees and coding trees
There is a bijection ∶  □ n,k →  n,k between the class  □ n,k of front-rooted Ω-k-trees with n hedra (where the root-front is labelled from 1 to k) and the class  n,k of all (k, Ω)-coding trees with n black vertices where the white root is labelled with the sequence (1, 2, … , k).
The correspondence is defined such that black nodes in a (k, Ω)-coding tree correspond to hedra in a Ω-k-tree. Every black node also gets a label which is equal to the label of one of the vertices of the corresponding hedron. A white node in a (k, Ω)-coding tree corresponds to a front of the Ω-k-trees and is labelled by the strictly increasing sequence (a 1 , a 2 , … , a k ) of labels of the corresponding front. A black node is a child of a white node if the corresponding hedron contains the corresponding front and the label of the black node is just the label of the vertex that is not contained in the front. Thus, if we start with the root-front of the Ω-k-tree, we can recursively build up a corresponding (k, Ω)-coding tree; see Figure 2.
With the help of this correspondence , the problem of counting Ω-k-trees with n hedra is reduced to counting the corresponding (k, Ω)-coding trees with n black nodes. In what follows we use the notation •-rooted (k, Ω)-coding trees if the white root has a fixed label and use the notation • − • (k, Ω)-coding tree if the white root of a reduced (k, Ω)-coding tree has a fixed label.
Let  □ k be the class of Ω-k-trees rooted at a fixed front {1, 2, … , k},  k be the class of (k, Ω)-coding trees, and  k be the class of • − • (k, Ω)-coding trees. Clearly the correspondence also establishes a bijection ∶  □ k →  k . The next goal is to formulate a recursive structure of  k and  k , where we make use of the standard construction rules for labelled combinatorial objects [28]. In this language every (k, Ω)-coding tree can be identified as a set of • − • (k, Ω)-coding trees with the outdegree set Ω: In terms of exponential generating functions (where the size is always the number of black nodes), we thus get We continue to decompose • − • (k, Ω)-coding tree. Let  • k be the class of •-rooted (k, Ω)-coding trees that are contained in the • − • (k, Ω)-coding trees. Then every tree from  • k can be identified as a set of • − • (k, Ω)-coding trees with the outdegree set Ω out = {i | i + 1 ∈ Ω, i ≥ 0} of the white nodes. Moreover, every • − • (k, Ω)-coding tree can be decomposed into a k-tuple of •-rooted (k, Ω)-coding trees. This yields the following specification: In terms of exponential generating functions, one gets In particular B k (z) satisfies By applying standard theory (see [28]) it follows that B k (z) has finite radius of convergence k,Ω and has also the property that B k ( k,Ω ) is finite. In particular the value of B k ( k,Ω ) is given by the relation In principle we have to distinguish between the aperiodic case, where all coefficients of B k (z) are positive and k,Ω is the only singularity on the radius of convergence |z| = k,Ω , and the periodic case, where B k (z) has several singularities on its radius |z| = k,Ω , see [9]. In the periodic case the coefficients [z n ]B k (z) are positive precisely for n = 1 mod , where = gcd(Ω out ). In both cases we obtain asymptotics for [z n ]B k (z), see Lemma 8 and Theorem 3 in [9]. So for the sake of simplicity, we assume that we are working in the aperiodic case. That is, [z n ]B k (z) has a unique dominant singularity z = k,Ω .
It follows immediately from (2) and (4) . One can analyze the asymptotic behaviors of b k,Ω (n) and c k,Ω (n) from the recursive equation (5), see [20,28]. This yields for some positive constants 1 , 2 . In order to obtain exact expressions for b k,Ω (n) we may apply the Lagrange inversion formula to Eq. 5. For the case Ω = N 0 this yields (8) and Hence the number of (k, Ω)-coding trees having an arbitrarily labelled root is given by which is consistent with the classical enumeration of k-trees [11,18,29,56].

Multitype Galton-Watson tree
In order to generate (uniformly) a • − • (k, Ω)-coding tree we make use of a multitype Galton-Watson process. Let • be an integer-valued random variable with probability distribution Furthermore let • be another integer-valued random variable with probability distribution The ( • , • )-multitype Galton-Watson tree  is then given by the family tree of a Galton-Watson branching process with alternating offspring distributions • and • . We start with a black node •, which gives birth to a set of white nodes • according to offspring distribution • , and each white node • gives birth to a set of black nodes • according to offspring distribution • . We denote by || the number of black nodes of .
Every ( • , • )-multitype Galton-Watson tree generated by this process is a plane tree, where the children of each node (white or black) are equipped with a left-to-right order. For simplicity, we assume that the white root of any • − • (k, Ω)-coding tree is always labelled with the sequence (1, 2, … , k).
In Lemma 3 we consider all different labellings of a ( • , • )-multitype Galton-Watson tree, where the labels on the white non-root nodes are determined by the labels on the black nodes and the white root. Two labellings are regarded as different if the edge sets are different.

Lemma 3
The following procedure terminates almost surely: Choose a uniform random labelling of  such that all black nodes of  are labelled with distinct integers (k + 1), (k + 2), … , (k + ||). 3. Add a white root to  and label it with the sequence (1, 2, … , k). This procedure draws a random • − • (k, Ω)-coding tree B k according to a critical Boltzmann distribution. That is, for every B ∈  n,k (of size n ≥ 1) we have Proof Note that different ( • , • )-multitype Galton-Watson trees after step (1) could lead to the same • − • (k, Ω)-coding tree after steps (2) and (3); see Figure 3. Let B be any • − • (k, Ω)-coding tree of size n and let T be the underlying unlabelled (k, Ω)-coding tree of B after we remove the white root and the edge connecting the white root.
We denote by (T) the set of embeddings of T into the plane where only the children of white nodes are equipped with a left-to-right order. Though every embedding of T is a ( • , • )-multitype Galton-Watson tree, only when applying the steps (2) and (3) on two different embeddings from the subset (T) can we possibly generate the same (k, Ω)-coding tree B; see Figure 3 for an example. The essential reason is the SET-relation in (3), where only the children of every white node in B are equipped with the SET-relation (without the left-to-right order).
Since the labels on all black nodes of T * determine the labels on the remaining white nodes, it suffices to count different labellings on black nodes. As we will show next, this leads us to the formula First there are n ways to label the black root of T * . Second, for the white node with outdegree 1 , there are ways to choose the labels on its black children, where we assume that T 1,1 , T 1,2 , … , T 1, 1 are distinct subtrees rooted at all these black children appearing with multiplicities m 1,1 , m 1,2 , … , m 1, 1 , respectively. Then m 1,1 + m 1,2 + · · · + m 1, 1 = 1 and there are ways to assign 1 labels to all these black children. We continue this process for the remaining white nodes with outdegrees 2 , … , kn and obtain  (1), which lead to the same • − • (k, Ω)-coding tree B after steps (2) and (3) in Lemma 3 subtrees rooted at each of the black children of a white node with outdegree i , which appear with ways to distribute i labels on the black children of a white node with outdegree i . We note that ∏ kn i=1 i is exactly the number |(T)| of different embeddings of T into the plane, where only the children of every white node are equipped with the left-to-right order. Thus, (12) follows.
For every T * ∈ (T) we choose a labelling of T * uniformly at random, and there is only one labelling of T * such that B is generated after step (3). That is, Consequently, the probability of drawing B of size n is given by This proves the lemma. ▪ Recall that by (1) every element of the class  k may be decomposed into a collection of elements from the class  k , where the number of components belongs to Ω. Let • denote a random integer from Ω with distribution given by for all i ∈ Ω. We are going to make use of the following critical Boltzmann sampling procedure for the class  k , which will be used in the proof of Theorem 2; see Section 4. It provides a coupling of a random (k, Ω)-coding tree with a forest of ( • , • )-multitype Galton-Watson trees. (For the reader's convenience we provide a self-contained proof, but the result may also be deduced from more general sampling principles given in [22].)

Lemma 4
The following procedure terminates almost surely: This procedure draws a random •-rooted (k, Ω)-coding tree C k according to a critical Boltzmann distribution. That is, for every C ∈  n,k (of size n) we have .
Proof For any •-rooted (k, Ω)-coding tree C ∈  n,k , assume that C can be decomposed into a set In other words, For any permutation = 1 2 · · · t ∈ S t we have , which leads to We may express the probability P[(B 1 , … , B t ) = (B 1 , … , B t )] by the product It follows from Lemma 3 that B i = B k is uniformly distributed among the | |L i |,k | many  k -objects with the label set L i for all i. Let us set i = |L i | for all 1 ≤ i ≤ t. Then Moreover, in view of Lemma 3, we may write Hence (15) now reads Combining this with (13) and (14) yields which completes the proof of the lemma. ▪ Note that • − • (k, Ω)-coding trees satisfy the specification (4), but the graph distance does not represent the distance relation in the corresponding Ω-k-trees; see Figure 2. Let B n,k denote a random • − • (k, Ω)-coding tree that is uniformly chosen from the class  n,k , that is, B n,k = (B k ∶ |B k | = n) where B k is a random (k, Ω)-coding tree that is generated by Lemma 3.

2.4
Ω-k-tree distance algorithm Let C n,k be a random (k, Ω)-coding tree and G □ n,k = −1 (C n,k ) be the corresponding Ω-k-tree under the inverse bijection −1 , where is given in section 2.2. In particular G □ n,k is rooted at the front [k]. We use the notation (i m , j k−m ) to represent a sequence of length k with m occurrences of i, followed by (k − m) occurrences of j.
The purpose of the following procedure is to determine the distances to vertex 1 in an Ω-k-tree G □ n,k . More precisely Darrasse and Soria [18] provided an algorithm for this task by marking the distances on the corresponding (k, Ω)-coding tree C n,k , which is similar to the algorithm given by Proskurowski in [59]. Note that every black node of the (k, Ω)-coding tree is related to a vertex of the corresponding Ω-k-tree via its label, and the vertices that label a white node of the (k, Ω)-coding tree represent k vertices that constitute a front of the corresponding Ω-k-tree.
Darrasse and Soria's algorithm runs as follows: Algorithm 1 Distances in an Ω-k-tree Input: a (k, Ω)-coding tree C and a sequence ( For example, we implement this algorithm on the (2, Ω)-coding tree (right) in Figure 2, which provides the distance of every black node to vertex 1 in Figure 4. The distance sequences on the white nodes help us to recursively mark the distances on the black nodes.
Remark 1 Based on this distance algorithm, Darrasse and Soria used a generating function approach to deduce a Rayleigh limiting distribution for the expected distances (scaled by √ n) between pairs of vertices in a random k-tree; see [18].

Gromov-Hausdorff-Prokhorov convergence and the CRT
We shall briefly recall some facts concerning the CRT and the Gromov-Hausdorff-Prokhorov metric.
We refer the reader to [2,15,26] and [55, Sec. 6] for more details. Let g ∶ [0, 1] → [0, ∞) be a non-negative continuous function such that g(0) = g(1) = 0. We may use it to define a pseudo-metric g by setting̃g for all u, v ∈ [0, 1]. Define an equivalence relation ∼ on [0, 1] by letting x ∼ y if̃g(x, y) = 0. Let T g = [0, 1]∕∼ and let g be the metric induced on T g bỹg. Then the associated metric space (T g , g ) is an R-tree (see [31]). We define the CRT to be the R-tree ( e , e ) encoded by the Brownian excursion of duration one e = (e(t), 0 ≤ t ≤ 1). The random metric space ( e , e ) is endowed with the probability measure e which is the push-forward of the Lebesgue measure on [0, 1]. We consider the CRT as a random point in the set K of all isometry-equivalence classes of compact measured metric spaces equipped with the Gromov-Hausdorff-Prokhorov distance GHP . We recall the definition of this metric, starting with the definition of Hausdorff distance.
Given two compact subsets K 1 , K 2 of a metric space (X, X ), their Hausdorff distance is The Prokhorov distance between two Borel probability measures 1 , 2 on X is defined by Let (X, 1 , 1 ), (Y, 2 , 2 ) be compact metric spaces equipped with Borel probability measures. For any metric space (E, E ) and isometric embeddings 1 ∶ X → E and 2 ∶ Y → E we may consider the push-forward measures 1 −1 1 and 2 −1 2 on E. The Gromov-Hausdorff-Prokhorov distance between these two spaces is given by where the infimum is taken over all isometric embeddings 1 , 2 from X, Y into any possible common space (E, E ). The Gromov-Hausdorff-Prokhorov distance satisfies the axioms of a premetric on the collection of compact metric spaces equipped with Borel probability measures. The corresponding metric on the quotient space K is complete and separable. That is, K is a Polish space. For simplicity we do not distinguish between a measured compact metric space and the corresponding equivalence class.
The main result on which the present work builds is the scaling limit for large Galton-Watson trees due to Aldous: Theorem 5 ([5]) Let T n be a critical Galton-Watson tree conditioned on having n vertices, where the offspring distribution has finite nonzero variance 2 . Let T n denote the uniform measure on the set of vertices of T n . Then, as n tends to infinity, T n with edges rescaled to length ∕(2 √ n) converges in distribution to the CRT, i.e., with respect to the Gromov-Hausdorff-Prokhorov distance.
Whenever there is no risk of confusion, we will write X, instead of (X, X , X ) for any scalar factor > 0 and any compact metric space (X, X ) that is equipped with a Borel probability measure X . Hence Theorem 5 may be stated by

Local convergence
Let  denote the collection of rooted graphs that are connected and locally finite. Given two rooted graphs G * = (G, v G ) and H * = (H, v H ) from , we define the distance where U m (G, v G ) denotes the rooted subgraph of G induced by all vertices with graph-distance at most m from the root v G , and U m (G, v G ) ≃ U m (H, v H ) denotes that the two subgraphs are isomorphic as rooted graphs. The distance d satisfies the axioms of a premetric and two elements from  have distance zero from each other if and only if they are isomorphic as rooted graphs. Hence d defines a complete and separable metric on the collection of all isomorphism classes of graphs from  [15,26]. A random rooted graph G * n = (G n , v n ) from  converges in the local weak sense towards a random element G * ∞ = (G ∞ , v ∞ ), denoted by if the corresponding isomorphism classes converge weakly with respect to the metric d. This is equivalent to requiring that for all fixed positive numbers r, and for all rooted graphs (G, v) it holds that

PROOF OF THEOREM 1
We recall that G n,k and G □ n,k = −1 (C n,k ) are identically distributed as random graphs. Hence it suffices to study the latter. We also recall C n,k denotes a random •-rooted (k, Ω)-coding tree of size n that is uniformly selected from the class  n,k , and that C n,k can be identified with a set of • − • (k, Ω)-coding trees, all of which have the same white root as C n,k .
We denote by L n,k one of the largest • − • (k, Ω)-coding trees that is contained in C n,k and denote by L n,k the size of L n,k . Equations (2) and (4) allow us to employ a unified analytic framework given by Xavier Gourdon [35]. As mentioned in section 2.2 we assume that we are in the aperiodic case, that is, we assume that z = k,Ω is the unique dominant singularity of B k (z). By general theory (see [20,28]) it follows that the singular expansion of B k (z) is given by where g(z), h(z) are analytic around z = k,Ω and z may vary in a Δ-domain of k,Ω for some > 0 and 0 < < ∕2. Furthermore, in view of (2), we can express C k (z) = F(B k (z)) where F(w) = ∑ i∈Ω w i (i!) −1 is an entire function. Thus, k,Ω is also the dominant singularity of C k (z). Hence, we are in the so-called non-critical case of [35]. By setting = 2, m = n − n 1∕2− , = 0 and = 1∕2 of Theorem 2 from [35], we have, for every (with 0 < < 1∕2) and for some constant c > 0 lim n→∞ P[L n,k ≤ n − n This implies that Hence in order to prove the Gromov-Hausdorff-Prokhorov scaling limit for G □ n = −1 (C n,k ) it suffices to prove such a limit for the randomly sized  k -object −1 (L n,k ). We know that L n,k conditioned on having a fixed size is distributed like a uniform  k -object of this size. As the random size L n,k tends weakly towards infinity, it is sufficient to show with respect to the Gromov-Hausdorff-Prokhorov distance. (Recall that G • n,k = −1 (B n,k ).) By Lemma 3 we may assume that B n,k = (B k ∶ |B k | = n). It is clear that any black node has k white children and the number of black children • of the white node in B n,k , other than the white root, follows the probability distribution (10). This implies that the number of black grandchildren •−• of any black node is the sum of k independent copies of • , that is,

So it satisfies
where E • = k −1 is obtained from (6) and (10), namely, For any two black nodes x, y in B n,k , we set For example, consider the (2, Ω)-coding tree in Figure 2. There we have n = 8 and k = 2. Let x, y be the black nodes labelled by 6,9 respectively. Then B 8,2 (x, y) = 2. For the case k ≠ 1, the distance B n,k (x, y) of two black nodes x, y in B n,k is different from the distance dist G • n,k (x, y) of x, y in the original Ω-k-tree G • n,k . In order to represent the distances dist G • n,k (x, y) for any two black nodes x, y in the tree B n,k , we need to decompose B n,k into blocks according to the distance table from Algorithm 1. We recall that Algorithm 1 marks every black node with a distance and every white node with a distance sequence.
With the help of these labels, we define (so-called) i-blocks (i = 1, 2, …) that decompose the random tree B n,k . A 1-block of B n,k is any subtree T of B n,k such that • T is rooted at the white root of B n,k and T is induced by this node and all the black descendants that are at distance one from vertex 1, while an i-block of B n,k , for i ≥ 2, is any subtree T of B n,k such that • T is rooted at a white node with distance sequence ((i − 1) k ) and T is induced by this node and all its black descendants that are at distance i from vertex 1.
Note that there is only one 1-block for any B n,k , but for i ≥ 2, B n,k could contain many i-blocks; see Figure 5. For any two black nodes x, y in B n,k , let where a is the minimal number of blocks necessary to cover the path connecting x and y. In particular if x, y are in the same block of B n,k , then B n,k (x, y) = 0. The following lemma will show that, for any two black nodes x, y, the distance dist G • n,k (x, y) is almost the same as the block-distance B n,k (x, y).

Lemma 6
Let B k be the random • − • (k, Ω)-coding tree given by Lemma 3 and let B n,k denote the tree B k conditioned on having n black nodes. Furthermore let G • n,k = −1 (B n,k ) be the corresponding Ω-k-tree of B n,k under the inverse bijection −1 . Then for any two black nodes x, y of G • n,k , ( 1 3 ) (0, 1 2 )

FIGURE 5
A decomposition of a random (3, Ω)-coding tree B n,3 into blocks (left) where the triple (a, b, c) of integers represents the distance sequence on the root of the block. A spine (right) consists of selected nodes in B n, 3 Proof If x, y are in the same block, i.e., B n,k (x, y) = 0. If both of them are in a 1-block, then If both of them are in an (i + 1)-block T for some i ≥ 1, recall that the root of any (i + 1)-block is a white node with distance sequence (i k ). We assume that the root of T has label (a 1 , a 2 , … , a k ). Then for x ∈ T, there exists an integer p such that dist G • n,k (a p , x) = 1. Otherwise if for all 1 ≤ m ≤ k we have dist G • n,k (a m , x) > 1. It follows that dist G • n,k (x, 1) > i + 1, which contradicts the fact that x is contained in an (i + 1)-block. Similarly, there is an integer q such that dist G • n,k (a q , y) = 1. Consequently which implies (23). If x, y are not in the same block, let b be the last common ancestor of x and y in B n,k and we suppose that b is a black node because the argument for the case when b is a white node follows analogously. Let a 1 (resp. b 1 ) be the second black node on the Then one of the shortest paths connecting x and y in G • n,k must pass node b, which implies that This is true because the Ω-k-tree corresponding to the subtree of B n,k rooted at a 1 and the Ω-k-tree corresponding to the subtree of B n,k rooted at b 1 are completely disjoint if we remove all (k−1) common vertices of 1 and 2 from G • n,k . These (k − 1) common vertices form a (k − 1)-clique and one of these common vertices is b. We shall show that B n,k (x, y) ≤ dist G • n,k (x, y). Note that and in the same way B n,k (y, b) ≤ dist G • n,k (y, b), which, together with (24), implies that k (x, y).
If x is contained in an (i + 1)-block, then there must exist a black node v 1 on the path b − • − a 1 − · · · − • − x, such that dist G • n,k (x, v 1 ) = 1 and v 1 is contained in an i-block. For the node v 1 , there exists a black node v 2 on the path such that v 2 is contained in an (i − 1)-block and dist G • n,k (x, v 2 ) = 2. We continue this process until we reach a black node v t such that v t and b are in the same block and k (x, b). Similarly, we can find a sequence of black nodes w 1 , … , w s from different blocks such that w s and b are in the same block and dist G • n,k (y, w s ) = s = B n,k (y, b). It follows that Since v t and w s are in the same block, we have shown that dist G • n,k (v t , w s ) ≤ 3, as desired. ▪ Lemma 6 allows us to transfer the distance dist G • n,k (x, y) of two vertices x, y in a random Ω-k-tree G • n,k to the distance B n,k (x, y) of two blocks in a random tree B n,k . In order to prove the convergence of G • n,k to the CRT  e , it is, thus, sufficient to prove that with high probability the difference between k B n,k (x, y) and B n,k (x, y) is uniformly small for all choices of x, y, where k is a properly chosen constant. For this purpose we consider the spine of a size-biased ( • , • )-multitype Galton-Watson tree. This construction is adapted from [1] and is a truncated version of the infinite size-biased Galton-Watson tree introduced by Kesten [47], Lyons and coworkers [52]. Furthermore, this idea has been used in studying the scaling limit of random graphs from subcritical graph classes [57] and was further generalized to the random -enriched trees [62].
Let̂• be a random variable with the size-biased distribution Note that this is a probability distribution on N + = {1, 2, …} becausê• ≥ 1 and E • = k −1 .
The size-biased ( • , • )-multitype Galton-Watson tree (m) , where m ≥ 1, is now defined as follows. It starts with a black-mutant node r which is a black node and it has white-node offspring according to • ; see (11). We now choose one of these white-node children uniformly at random and call it white-mutant. Every other white-node offspring is called white-normal. Each white-mutant node has black-node offspring according tô•, while each white-normal node has black-node offspring according to • (see (10)) and all these black-node offspring are called black-normal. We again choose one of the black-node children of the white-mutant node uniformly at random and call it heir. All other black-node children are also called black-normal, where all black-normal nodes have white-normal children according to independent copies of • . The heir is black-mutant if it has depth less than 2m, and we proceed recursively as long as the heir is black-mutant. Finally if the heir has depth 2m, then all black nodes at this level are normal, which again have white-normal offspring according to independent copies of • .
Let h m denote the heir that is the (m + 1)th black-mutant node of (m) , then the path from the black root r to h m is called a spine of (m) . For simplicity, we adopt the notations of B, T, T * , (T), (T * ) and outdegree sequence ( 1 , 2 , … , kn ) of T from the proof of Lemma 3. Our aim is to show that for every T * ∈ (T) and every fixed spine of length 2m that connects the root and v of T * we have The probability that a given black-mutant node has one white-mutant child and this white-mutant child has q black-node children where one of them is chosen as heir is, which implies (26). We recall that B k denotes a random (k, Ω)-coding tree that is generated in Lemma 3. Our next claim is whereB (m) k is a size-biased (k, Ω)-coding tree that is constructed as follows: 1. Draw a sized-biased ( • , • )-multitype Galton-Watson tree (m) . 2. Choose a uniform random labelling of (m) such that all black nodes of (m) are labelled with distinct integers (k + 1), (k + 2), … , (k + | (m) |). 3. Add a white root to (m) and label it with (1, 2, … , k).
This construction is analogous to the one in Lemma 3, which, in combination with (26), implies that That is, (27) is true. This relation shows, once the spine is fixed, that the probability that the size-biased treeB (m) k equals B is the same as the probability of the event B k = B (see equation (3.2) from [1] for a size-biased Galton-Watson tree and Figure 6 for an example).

Lemma 7
Let  n,k be the class of • − • (k, Ω)-coding trees of size n such that the white root has label (1, 2, … , k) and let B n,k ∈  n,k be selected uniformly at random. Let k = kH k and let r be the only black-node child of the white root of B n,k . Then for all 0 < s, < 1∕2 and for every black node v ∈ B n,k such that B n,k (r, v) ≥ n s , the property holds with probability tending to 1 as n tends to infinity.
Proof Suppose that the opposite of (28) is true, i.e., there exists v * ∈ B k such that For any such B k , fix one black node v * ∈ B k having this property. Furthermore let  1 be the collection of all such pairs (B k , v * ). We are going to show that With the help of Lemma 3 and by applying [20,Theorem 2.19]) we obtain, with 2 Ω = kVar • , the asymptotic relation A (3, Ω)-coding tree B 3 (left) and a size-biased oneB (m) 3 with a spine consisting of all black-mutant nodes and white-mutant nodes (right), where every black node is marked with a distance and every white node is marked with a distance sequence Consequently, we have Furthermore, Now for any (B, v) ∈  1 we consider the spine of B that ends at v. Then from (27) we see that where p m denotes the probability, that the block-distance m of h m to r in the random size-biased (k, Ω)-coding treeB (m) k satisfies | k m − m| > m.
Set i = i + 1, where i equals the graph-distance of h i to vertex 1 in the corresponding Ω-k-tree −1 (B (m) k ). In other words, after we implement Algorithm 1 onB (m) k , the distance marked on h i is equal to i .
From Algorithm 1 we observe that i+1 only depends on i and the distance sequence of the white node that has h i as a child. Let X i ≥ 1 be the number of integers contained in this distance sequence that equal i − 1, as a result, (X i , i ) i is a Markov chain that starts at (X 0 , 0 ) = (1, 1) where X 0 = 1 counts the number of zeros in the distance sequence of the white root (0, 1 k−1 ) and 0 = 1 is the distance marked on r after using Algorithm 1. Here, for all i ≥ 1, is distributed as follows. In particular, (X i ) i is a Markov chain with state space {1, 2, … , k} and transition matrix The stationary distribution = ( i ) k i=1 satisfies = P, which yields Since (X m , m ) is a Markov-additive process, we apply a general large deviation result for these processes [51,Thm. 3.3] and obtain that for every * > 0, there is a constant c( * ) > 0 such that holds for all m, n s ≤ m ≤ n, where = 1 ∕k = (kH k ) −1 = −1 k is the mean of a typical step of the chain in the stationary state. This implies that p m is exponentially small. Hence, from (33) we conclude that P[(B k , v * ) ∈  1 | | |B k | = n] → 0 as n → ∞, which completes the proof of the lemma. ▪ Now we are ready to prove our first main result.
Let S n be a critical Galton-Watson tree conditioned on having size n, where the number of offspring follows the distribution of the random variable •−• . Then by applying Theorem 5 we obtain In order to prove Theorem 1, it suffices to show (19), as we have argued in section 2.1 and Section 3. In view of (G • n,k ) = B n,k , Lemma 6 and (34), it is also sufficient to prove Let r be the only black-node child of the white root of B n,k , then for any two black nodes u, v of B n,k , let o be the last common ancestor of u, v. Without loss of generality, suppose that o is a black node, then one can easily check that which is also true if B n,k is replaced by B n,k . Thus, in order to show (35), we only need to prove Since B n,k (r, v) ≤ B n,k (r, v), we find that (36) is true for all v ∈ B n,k such that B n,k (r, v) < n s for 0 < s < 1∕2. So we only need to consider vertices v ∈ B n,k such that B n,k (r, v) ≥ n s .
It follows from Lemma 7 that holds with probability tending to 1 as n tends to infinity, where H(S n ) is the height of S n . By applying the tail-bounds and the left-tail upper bounds for the height H(S n ) of S n (see [1]), we conclude that H(S n )∕ √ n tends to zero as n tends to infinity. In consequence, (36) is true. This finally completes the proof of Theorem 1.

PROOF OF THEOREM 2
In this section, we are going to construct an infinite Ω-k-tree G ∞,k that is rooted at a front of distinguishable vertices. We then establish the convergence of G • n,k toward this random graph in the sense, that for each fixed integer ≥ 0 the front-rooted sub-Ω-k-tree U (G • n,k ) that is induced by all vertices at ψ FIGURE 7 Construction of (k, Ω)-coding trees out of plane trees where the outdegree of each vertex is a multiple of k, illustrated for the special case k = 2 distance at most from the marked front, converges in distribution to the corresponding sub-Ω-k-tree U (G ∞,k ) of the limiting object.
By the discussion in section 2.1, the random Ω-k-tree G • n,k is up to relabelling distributed like the Ω-k-tree G □ n,k that is rooted at a fixed front with labels from 1 to k. Hence we only need to study the neighborhoods of the root-front. If we distinguish any fixed vertex of the marked front in G □ n,k , for example the vertex with label 1, and also distinguish a fixed vertex of the marked front in G ∞,k , then our limit may be interpreted as a classical local weak limit of a sequence of vertex-rooted random graphs as discussed in section 2.6. This may be justified by the following two arguments. First, as rooted graphs, all k possible vertex-rootings of G □ n,k are identically distributed, and we shall see below that the same is true for the limit G ∞,k . Second, the -neighborhood of a vertex of any front-rooted Ω-k-tree is always a subgraph of the -neighborhood of the marked front, and hence weak convergence of the neighborhoods of the front implies weak convergence of the neighborhoods of the vertices.
The strategy of the proof is as follows. We generate a random Ω-k-tree G □ n,k by applying the bijection −1 ∶  n,k →  □ n,k to the random (k, Ω)-coding tree C n,k . This random coding tree may be generated by conditioning C k on having n black nodes (as done in Lemma 4). We observe that any ordered tree of white vertices where the outdegree of every vertex is a multiple of k may be interpreted as a (k, Ω)-coding tree by adding black vertices in a canonical way. Here different plane trees correspond to the same unlabelled (k, Ω)-tree, but this will not be an issue. We use this construction in order to formulate a coupling of the random (k, Ω)-coding tree C k with a Galton-Watson tree T • that has a modified root-degree. If we condition this locally modified Galton-Watson tree on having (kn + 1) vertices, then the result T n,• corresponds, up to relabelling, to the (k, Ω)-coding tree C n,k . By the same kind of arguments that provide local convergence of simply generated trees, we obtain that the random tree T n,• converges weakly toward an infinite plane tree T ∞,• , which may be interpreted as a (k, Ω)-coding tree C ∞,k , and consequently also as a front-rooted Ω-k-tree G ∞,k . The final step in the proof is to deduce local convergence of a random Ω-k-tree G □ n,k from this convergence of random trees.
The construction of a (k, Ω)-coding tree (T) out of a plane tree T, where the outdegree of each vertex is a multiple of k, is straight-forward. We canonically partition the offspring set of each vertex v of T into an ordered list of groups G 1 (v), G 2 (v), … of k consecutive vertices. The edges between v and its offspring are then deleted, and for each group G i (v), we add a black offspring vertex u i (v) to v and add further edges such that G i (v) is the offspring set of u i (v). This construction is illustrated in Figure 7.
We may now use this to formulate a coupling of a critical Boltzmann distributed (k, Ω)-coding tree with a modified Galton-Watson tree. Recall that in Lemma 4 we constructed a random (k, Ω)-coding tree C k by starting with a white root, and connecting it with the roots of a random number • of independent • − • (k, Ω)-coding trees, where each of them is sampled independently according to the random tree . Recall that the tree  starts with a black node with k white nodes as offspring. Each of the white nodes receives a random number • of black nodes, which follows the distribution given in (10). Then the sampler is applied recursively. That is, every black node receives k white nodes as offspring and each of them receives a random number of black offspring, and so on.
Let T • denote a modified Galton-Watson tree, where each vertex receives offspring according to an independent copy of ∶= k • , except for the root, which receives offspring according to ∶= k • . The order in which the recursion takes place in  does not matter, hence the (k, Ω)-coding tree (T • ) is up to relabelling distributed like a (k, Ω)-coding tree C k . Moreover, let T n,• denote the tree T • conditioned on having (kn + 1) vertices, then (T n,• ) is distributed as the random (k, Ω)-coding tree C n,k .
Note that (21) implies that E = 1, and both and have finite exponential moments. We define the size-biased versions of these offspring distributions by Let T ∞,• denote the following random infinite (but locally finite) plane tree. There are two types of non-root vertices, mutant and normal. The root receives offspring according tô, and one of its sons is selected uniformly at random and declared mutant, whereas the others are normal. Normal vertices receive offspring according to an independent copy of , all of which are normal. Mutant vertices receive offspring according to an independent copy of̂, and among them one is selected uniformly at random and declared mutant, whereas the others are normal. Hence T ∞,• is an infinite plane tree with a distinguished path that starts at the root and traverses the mutant vertices. We call this path the spine of T ∞,• .
We describe the convergence of the random tree T n,• toward the limit tree T ∞,• using a slight modification of the arguments in Janson's survey [42]. For each plane tree T and each integer h ≥ 0 let T [h] denote the tree obtained by cutting away all vertices at height larger than h.
As T ∞,• has infinite height, this already implies that H(T n,• ) ≥ h occurs with probability tending to 1, and consequently T A classical combinatorial observation, also called the cycle lemma, states that for any sequence x 1 , … , x s ≥ −1 of integers satisfying ∑ s i=1 x i = −r for some r ≥ 1, there are precisely r integers 1 ≤ u ≤ s such that the cyclically shifted sequence x (u) i = x 1+(i+u) mod s satisfies ∑ i=1 x (u) i > r for all 1 ≤ ≤ s − 1; see for example [42,Lem. 15.3]. Consequently, (38) may be simplified to The tree T has precisely (D − t + 1) vertices at height h. Hence the event T [h] ∞,• = T corresponds to precisely (D − t + 1) possible outcomes for the first (h + 1) levels of T ∞,• , depending on the location for the unique spine vertex at height h. Each outcome has the same probability Let (o) denote the root-degree of T • . It holds, since has finite exponential moments, that P[ ≥ log(n) 2 ] is exponentially small. Hence, using the cycle lemma and the central local limit theorem in an identical fashion as above, it follows that which, together with (40), implies (37), which completes the proof. ▪ We are now finally in the position to complete the proof of our second main theorem.
Proof of Theorem 2 Let be an integer and let G be an arbitrary finite unlabelled Ω-k-tree that is rooted at a front. We claim that there exist an integer L ≥ 0, that depends on both and G, and a set  of finite plane trees, such that any plane tree T, that corresponds to a (k, Ω)-coding tree (T) and hence to a front-rooted Ω-k-tree G(T) ∶= −1 ( (T)), has the property U (G(T)) = G if and only if T [L] ∈ . Before we prove the claim, we show that Theorem 2 is a direct consequence of this claim. By where G ∞,k denotes the Ω-k-tree corresponding to T ∞,• , as desired.
In order to prove the existence of such an integer L and a set  we argue as follows. To each plane tree T we associate a unique sequence of increasing subtrees T 0 , T 1 , … of T that all contain the root-vertex of T and have the property G(T i ) = U i (G(T)) for all i. Of course, the tree T may, in general, have arbitrarily large height. However, in order to satisfy G(T ) = G, the tree T may not have more vertices than the number of fronts in G. In particular, the height of T is bounded by the number of fronts of G. Hence there exists a finite integer L such that for every plane tree T we may decide whether U (G(T)) = G is true or not just by looking at T [L] . ▪