The distribution of the size of the ancestor‐tree and of the induced spanning subtree for random trees

We consider two tree statistics that extend in a natural way the parameters depth of a node resp. distance between two nodes. The ancestor‐tree of p given nodes in a rooted tree T is the subtree of T, spanned by the root and these p nodes and generalizes the depth (ancestor‐tree of a single node), whereas the spanning subtree induced by p given nodes in a tree T generalizes the distance (induced spanning subtree of two nodes). We study the random variables size of the ancestor‐tree resp. spanning subtree size for two tree families, the simply generated trees and the recursive trees. We will assume here the random tree model and also that all () possibilities of selecting p nodes in a tree of size n are equally likely. For random simply generated trees we can then characterize for a fixed number p of chosen nodes the limiting distribution of both parameters as generalized Gamma distributions, where we prove the convergence of the moments too. For some specific simply generated tree families we can give exact formulæ for the first moments. In the instance of random recursive trees, we will show that the considered parameters are asymptotically normally distributed, where we can give also exact formulæ for the expectation and the variance. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Level of nodes in increasing trees revisited

Simply generated families of trees are described by the equation T(z) = ϕ(T(z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ∈ { 1,…,n}, no label occurs twice, and whenever we proceed from the root to a leaf, the labels are increasing. This leads to the concept of simple families of increasing trees. Three such families are especially important: recursive trees, heap ordered trees, and binary increasing trees. They belong to the subclass of very simple families of increasing trees, which can be characterized in 3 different ways. This paper contains results about these families as well as about polynomial families (the function ϕ(u) is just a polynomial). The random variable of interest is the level of the node (labelled) j, in random trees of size n ≥ j. For very simple families, this is independent of n, and the limiting distribution is Gaussian. For polynomial families, we can prove this as well for j,n → ∞ such that n − j is fixed. Additional results are also given. These results follow from the study of certain trivariate generating functions and Hwang's quasi power theorem. They unify and extend earlier results by Devroye, Mahmoud, and others. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Cutting down very simple trees

We study here, by using a recursive approach, the number of random cuts that are necessary to destroy a random tree of size n for simply generated tree families. Crucial for the applicability of such a recursive approach is a "randomness-preservation" property when cutting off a random edge. We can fully characterize the subclass of simply generated tree families, which satisfy this property and show then for for all these tree families that the number of random cuts to destroy a random size-n tree is asymptotically, for n → ∞, Rayleigh distributed.     

On the degree distribution of the nodes in increasing trees

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i.e., labellings of the nodes by distinct integers of the set {1,…,n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recursive trees, plane-oriented recursive trees and binary increasing trees. We study the quantity degree of node j in a random tree of size n and give closed formulae for the probability distribution and all factorial moments for those subclass of tree families, which can be constructed via a tree evolution process. Furthermore limiting distribution results of this parameter are given, which completely characterize the phase change behavior depending on the growth of j compared to n.     

Left and Right Length of Paths in Binary Trees or on a Question of Knuth

We consider extended binary trees and study the joint right and left depth of leaf j, where the leaves are labelled from left to right by 0, 1, . . . , n, and the joint right and left external pathlength of binary trees of size n. Under the random tree model, i.e., the Catalan model, we characterize the joint limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf j in a random size-n binary tree when j ~ ρn with 0 < ρ > 1, as well as the joint limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random size-n binary tree. This work was supported by the Austrian Science Foundation FWF, grant S9608-N13.     

The cut‐tree of large trees with small heights

We destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. Informally, the associated cut‐tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut‐tree in the Gromov‐Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has branching points close to the root and height of order . In particular, we consider uniform random recursive trees, binary search trees, scale‐free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin (Probab Stat 5 (2015), 478–488) for the cut‐tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer (Online J Anal Combin (2014), 26) on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut‐tree of other classes of trees. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 404–427, 2017     

Depth Properties of scaled attachment random recursive trees

We study depth properties of a general class of random recursive trees where each node i attaches to the random node \begin{align*}\left\lfloor iX_i\right\rfloor\end{align*} and X₀,…,X_n is a sequence of i.i.d. random variables taking values in [0,1). We call such trees scaled attachment random recursive trees (sarrt). We prove that the typical depth D_n, the maximum depth (or height) H_n and the minimum depth M_n of a sarrt are asymptotically given by D_n ～μ^‐1 log n, H_n ～ α_max log n and M_n ～ α_min log n where μ,α_max and α_min are constants depending only on the distribution of X₀ whenever X₀ has a density. In particular, this gives a new elementary proof for the height of uniform random recursive trees H_n ～ elog n that does not use branching random walks.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Recursive functions on conditional Galton‐Watson trees

A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random element U. The value of the root is the key quantity of interest in general. In this study, all node values and function values are in a finite set S. In this note, we describe the limit behavior when the leaf values are drawn independently from a fixed distribution on S, and the tree T_n is a random Galton‐Watson tree of size n.     

Limiting Distributions for the Number of Inversions in Labelled Tree Families

We consider the so-called simple families of labelled trees, which contain, e.g., ordered, unordered, binary, and cyclic labelled trees as special instances, and study the global and local behaviour of the number of inversions. In particular, we obtain limiting distribution results for the total number of inversions as well as the number of inversions induced by the node labelled j in a random tree of size n.     

Limit laws for local counters in random binary search trees

Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for w-dependent random variables. Examples include: the number of leaves (L_n), the number of nodes with k descendants (k fixed), the number of nodes with no left child, the number of nodes with k left descendants. Some of these results can also be obtained via the theory of urn models, but the present method seems easier to apply.     

Expectation transfer between branching processes and random trees

An approach for translating results on expected parameter values from subcritical Galton–Watson branching processes to simply generated random trees under the uniform model is outlined. As an auxiliary technique for asymptotic evaluations, we use Flajolet's and Odlyzko's transfer theorems. Some classical results on random trees are re-derived by the mentioned approach, and some new results are presented. For example, the asymptotic behavior of linearly recursive tree parameters is described and the asymptotic probability of level k to contain exactly one node is determined. © 1993 John Wiley & Sons, Inc.     

Nodes of large degree in random trees and forests

We study the asymptotic behavior of the number N_k,n of nodes of given degree k in unlabeled random trees, when the tree size n and the node degree k both tend to infinity. It is shown that N_k,n is asymptotically normal if and asymptotically Poisson distributed if . If , then the distribution degenerates. The same holds for rooted, unlabeled trees and forests. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Graph limits of random graphs from a subset of connected k‐trees

For any set Ω of non‐negative integers such that , we consider a random Ω‐k‐tree G_n,k that is uniformly selected from all connected k‐trees of (n + k) vertices such that the number of (k + 1)‐cliques that contain any fixed k‐clique belongs to Ω. We prove that G_n,k, scaled by where H_k is the kth harmonic number and σ_Ω > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω‐k‐tree to an infinite but locally finite random Ω‐k‐tree G_∞,k.     

The n‐ordered graphs: A new graph class

For a positive integer n, we introduce the new graph class of n‐ordered graphs, which generalize partial n‐trees. Several characterizations are given for the finite n‐ordered graphs, including one via a combinatorial game. We introduce new countably infinite graphs R⁽ⁿ⁾, which we name the infinite random n‐ordered graphs. The graphs R⁽ⁿ⁾ play a crucial role in the theory of n‐ordered graphs, and are inspired by recent research on the web graph and the infinite random graph. We characterize R⁽ⁿ⁾ as a limit of a random process, and via an adjacency property and a certain folding operation. We prove that the induced subgraphs of R⁽ⁿ⁾ are exactly the countable n‐ordered graphs. We show that all countable groups embed in the automorphism group of R⁽ⁿ⁾. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 204–218, 2009     

Covering minimum spanning trees of random subgraphs

We consider the problem of finding a sparse set of edges containing the minimum spanning tree (MST) of a random subgraph of G with high probability. The two random models that we consider are subgraphs induced by a random subset of vertices, each vertex included independently with probability p, and subgraphs generated as a random subset of edges, each edge with probability p. Let n denote the number of vertices, choose p ∈ (0, 1) possibly depending on n, and let b = 1/(1 ? p). We show that in both random models, for any weighted graph G, there is a set of edges Q of cardinality O(n log_bn) that contains the minimum spanning tree of a random subgraph of G with high probability. This result is asymptotically optimal. As a consequence, we also give a bound of O(kn) on the size of the union of all minimum spanning trees of G with some k vertices (or edges) removed. More generally, we show a bound of O(n log_bn) on the size of a covering set in a matroid of rank n, which contains the minimum‐weight basis of a random subset with high probability. Also, we give a randomized algorithm that calls an MST subroutine only a polylogarithmic number of times and finds the covering set with high probability. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

The continuum random tree is the scaling limit of unlabeled unrooted trees

We show that the uniform unlabeled unrooted tree with n vertices and vertex degrees in a fixed set converges in the Gromov‐Hausdorff sense after a suitable rescaling to the Brownian continuum random tree. This confirms a conjecture by Aldous (1991). We also establish Benjamini‐Schramm convergence of this model of random trees and provide a general approximation result, that allows for a transfer of a wide range of asymptotic properties of extremal and additive graph parameters from Pólya trees to unrooted trees.     

Projections of the Aldous chain on binary trees: Intertwining and consistency

Consider the Aldous Markov chain on the space of rooted binary trees with n labeled leaves in which at each transition a uniform random leaf is deleted and reattached to a uniform random edge. Now, fix 1 ≤ k<n and project the leaf mass onto the subtree spanned by the first k leaves. This yields a binary tree with edge weights that we call a “decorated k‐tree with total mass n.” We introduce label swapping dynamics for the Aldous chain so that, when it runs in stationarity, the decorated k‐trees evolve as Markov chains themselves, and are projectively consistent over k. The construction of projectively consistent chains is a crucial step in the construction of the Aldous diffusion on continuum trees by the present authors, which is the n→∞ continuum analog of the Aldous chain and will be taken up elsewhere.     

Isolating a Leaf in Rooted Trees via Random Cuttings

We consider a recursive procedure for destroying rooted trees and isolating a leaf by removing a random edge and keeping the subtree, which does not contain the original root. For two tree families, the simply generated tree families and increasing tree families, we study here the number of random cuts that are necessary to isolate a leaf. We can show limiting distribution results of this parameter for simply generated trees and certain increasing trees. This work was supported by the Austrian Science Foundation FWF, grant S9608-N13.     

The shape of large Galton‐Watson trees with possibly infinite variance

Let T be a critical or subcritical Galton‐Watson family tree with possibly infinite variance. We are interested in the shape of T conditioned to have a large total number of vertices. For this purpose we study random trees whose conditional distribution given their size is the same as the respective conditional distribution of T. These random family trees have a simple probabilistic structure if decomposed along the lines of descent of a number of distinguished vertices chosen uniformly at random. The shape of the subtrees spanned by the selected vertices and the root depends essentially on the tail of the offspring distribution: While in the finite variance case the subtrees are asymptotically binary, other shapes do persist in the limit if the variance is infinite. In fact, we show that these subtrees are Galton‐Watson trees conditioned on their total number of leaves. The rescaled total size of the trees is shown to have a gamma limit law. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004