The shape of random tanglegrams

A tanglegram consists of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the trees. We show that the two halves of a random tanglegram essentially look like two independently chosen random plane binary trees. This fact is used to derive a number of results on the shape of random tanglegrams, including theorems on the number of cherries and generally occurrences of subtrees, the root branches, the number of automorphisms, and the height. For each of these, we obtain limiting probabilities or distributions. Finally, we investigate the number of matched cherries, for which the limiting distribution is identified as well.     

Profiles of random trees: Plane‐oriented recursive trees

We derive several limit results for the profile of random plane‐oriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width, and the correlation coefficients of two level sizes. Most of our proofs are based on a method of moments. We also discover an unexpected connection between the profile of plane‐oriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size). © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

The height of multiple edge plane trees

Multi-edge trees as introduced in a recent paper of Dziemiańczuk are plane trees where multiple edges are allowed. We first show that d-ary multi-edge trees where the out-degrees are bounded by d are in bijection with classical d-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically.     

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.     

On the internal path length of d‐dimensional quad trees

It is proved that the internal path length of a d‐dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the m‐ary search trees. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 5: 25–41, 1999     

Analysis of three graph parameters for random trees

We consider three basic graph parameters, the node‐independence number, the path node‐covering number, and the size of the kernel, and study their distributional behavior for an important class of random tree models, namely the class of simply generated trees, which contains, e.g., binary trees, rooted labeled trees, and planted plane trees, as special instances. We can show for simply generated tree families that the mean and the variance of each of the three parameters under consideration behave for a randomly chosen tree of size n asymptotically ～μn and ～νn, where the constants μ and ν depend on the tree family and the parameter studied. Furthermore we show for all parameters, suitably normalized, convergence in distribution to a Gaussian distributed random variable. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

On plane trees with a prescribed number of valency set realizations

We describe valency sets of plane bicolored trees with a prescribed number of realizations by plane trees. Three special types of plane trees are defined: chains, trees of diameter 4, and special trees of diameter 6. We prove that there is a finite number of valency sets that have R realizations as plane trees and do not belong to these special types. The number of edges of such trees is less than or equal to 12R  + 2. The complete lists of valency sets of plane bicolored trees with 1, 2, or 3 realizations are presented. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 9–17, 2007.     

The distribution of nodes of given degree in random trees

Let 𝒯_n denote the set of unrooted unlabeled trees of size n and let k ≥ 1 be given. By assuming that every tree of 𝒯_n is equally likely, it is shown that the limiting distribution of the number of nodes of degree k is normal with mean value ∼ μ_kn and variance ∼ σn with positive constants μ_k and σ_k. Besides, the asymptotic behavior of μ_k and σ_k for k → ∞ as well as the corresponding multivariate distributions are derived. Furthermore, similar results can be proved for plane trees, for labeled trees, and for forests. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 227–253, 1999     

The Height of Increasing Trees

Increasing trees have been introduced by Bergeron, Flajolet, and Salvy [1]. This kind of notion covers several well-know classes of random trees like binary search trees, recursive trees, and plane oriented (or heap ordered) trees. We consider the height of increasing trees and prove for several classes of trees (including the above mentioned ones) that the height satisfies EH _n ~ γlogn (for some constant γ > 0) and Var H _n =  O(1) as n → ∞. The methods used are based on generating functions. This research was supported by the Austrian Science Foundation FWF, project S9604, that is part of the Austrian National Research Network "Analytic Combinatorics and Probabilistic Number Theory".     

Thermodynamic limit for large random trees

We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We compute the limiting distribution explicitly and study its properties. We introduce an infinite random tree consistent with these limiting distributions and show that it satisfies a certain form of the Markov property. We also study the growth of this tree and prove several limit theorems including a diffusion approximation. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

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     

Achiral plane trees

Harary and Robinson showed that the number a_n of achiral planted plane trees with n points coincides with the number p_n of achiral plane trees with n points, for n ? 2. They posed the problem of finding a natural structural correspondence which explains this coincidence. In the present paper this problem is answered by constructing two-to-one correspondences from certain sets of binary sequences to each of the sets of trees concerned, giving a structural basis for the equation 2a_n = 2p_n. Answers are also supplied to similar correspondence-type problems of Harary and Robinson, concerning planted plane trees, and achiral rooted plane trees. In addition, each of these four types of plane trees are counted with numbers of points and endpoints as the enumeration parameters. The results all show a symmetry with respect to the number of endpoints which is not shared by the set of all plane trees.     

Probabilistic and fractal aspects of Lévy trees

We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted -trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the well-known property for Galton-Watson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree. We finally discuss some applications to super-Brownian motion with a general branching mechanism.     

On edge-weighted recursive trees and inversions in random permutations

We introduce random recursive trees, where deterministically weights are attached to the edges according to the labeling of the trees. We will give a bijection between recursive trees and permutations, which relates the arising edge-weights in recursive trees with inversions of the corresponding permutations. Using this bijection we obtain exact and limiting distribution results for the number of permutations of size n, where exactly m elements have j inversions. Furthermore we analyze the distribution of the sum of labels of the elements, which have exactly j inversions, where we can identify Dickman's infinitely divisible distribution as the limit law. Moreover we give a distributional analysis of weighted depths and weighted distances in edge-weighted recursive trees.     

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     

On generalized independent subsets of trees

A natural generalization of the widely discussed independent (or “internally stable”) subsets of graphs is to consider subsets of vertices where no two elements have distance less or equal to a fixed number k (“k-independent subsets”). In this paper we give asymptotic results on the average number of ?-independent subsets for trees of size n, where the trees are taken from a so-called simply generated family. This covers a lot of interesting examples like binary trees, general planted plane trees, and others.     

The butterfly decomposition of plane trees

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition, which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem as well as some generalizations and variations follow quickly from the butterfly decomposition. We next obtain two involutions on free Dyck paths and free Schröder paths, leading to parity results and combinatorial identities. We also use the butterfly decomposition to give a combinatorial treatment of Klazar's generating function for the number of chains in plane trees. Finally we study the total size of chains in plane trees with n edges and show that the average size of such chains tends asymptotically to (n+9)/6.     

The Variance of Level Numbers in Certain Families of Trees

For the families of t-ary trees and planted plane trees, the variance of the level numbers (i.e., the depths of specified endnodes) is determined. We apply different types of techniques, such as singularity analysis (Kirschenhofer's diagonalization method) or a probabilistic approach using weak convergence and uniform integrability. In the case of binary trees, an exact variance formula for finite tree size is established; in the other cases, asymptotic equivalents are derived. Also some results on higher moments are presented.     

Noncrossing trees are almost conditioned Galton–Watson trees

A noncrossing tree (NC‐tree) is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments that do not cross. In this article, we show that NC‐trees with size n are conditioned Galton–Watson trees. As corollaries, we give the limit law of depth‐first traversal processes and the limit profile of NC‐trees. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 115–125, 2002     

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.