An involution on increasing trees

We introduce a method to construct bijections on increasing trees. Using this method, we construct an involution on increasing trees, from which we obtain the equidistribution of the statistics ‘number of odd vertices’ and ‘number of even vertices at odd levels’. As an application, we deduce that the expected value of the number of even vertices is twice the expected value of the number of odd vertices in a random recursive tree of given size.     

The structure and distances in Yule recursive trees

Based on uniform recursive trees, we introduce random trees with the factor of time, which are named Yule recursive trees. The structure and the distance between the vertices in Yule recursive trees are investigated in this paper. For arbitrary time t > 0, we first give the probability that a Yule recursive tree Y_t is isomorphic to a given rooted tree γ; and then prove that the asymptotic distribution of ζ_t,m, the number of the branches of size m, is the Poisson distribution with parameter λ = 1/m. Finally, two types of distance between vertices in Yule recursive trees are studied, and some limit theorems for them are established.© 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

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     

An analysis of the size of the minimum dominating sets in random recursive trees, using the Cockayne-Goodman-Hedetniemi algorithm

A random recursive tree on n vertices is either a single isolated vertex (for n=1) or is a vertex v_n connected to a vertex chosen uniformly at random from a random recursive tree on n−1 vertices. Such trees have been studied before [R. Smythe, H. Mahmoud, A survey of recursive trees, Theory of Probability and Mathematical Statistics 51 (1996) 1-29] as models of boolean circuits. More recently, Barabási and Albert [A. Barabási, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509-512] have used modifications of such models to model for the web and other “power-law” networks.A minimum (cardinality) dominating set in a tree can be found in linear time using the algorithm of Cockayne et al. [E. Cockayne, S. Goodman, S. Hedetniemi, A linear algorithm for the domination number of a tree, Information Processing Letters 4 (1975) 41-44]. We prove that there exists a constant d?0.3745… such that the size of a minimum dominating set in a random recursive tree on n vertices is dn+o(n) with probability approaching one as n tends to infinity. The result is obtained by analysing the algorithm of Cockayne, Goodman and Hedetniemi.     

Depth in bucket recursive trees with variable capacities of buckets

We consider bucket recursive trees of sizen consisting of all buckets with variable capacities1,2,...,b and with a specifc stochastic growth rule.This model can be considered as a generalization of random recursive trees like bucket recursive trees introduced by Mahmoud and Smythe where all buckets have the same capacities.In this work,we provide a combinatorial analysis of these trees where the generating function of the total weights satisfes an autonomous frst order diferential equation.We study the depth of the largest label（i.e.,the number of edges from the root node to the node containing label n）and give a closed formula for the probability distribution.Also we prove a limit law for this quantity which is a direct application of quasi power theorem and compute its mean and variance.Our results for b=1 reduce to the previous results for random recursive trees.     

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     

The Degree Profile in Some Classes of Random Graphs that Generalize Recursive Trees

We study the degree profile for a number of classes of random graphs that arise as generalizations of recursive trees, including random circuits and random recursive trees endowed with the power of choice. We investigate the distribution of the degrees of nodes that appear in various stages of the insertion process in each of these graph types. For these classes, we will see phase transitions in degrees depending on the stage—early stages are associated with normal distributions, intermediate stages are associated with the Poisson distribution and in the late stages the degrees become degenerate.     

A family of random trees with random edge lengths

We introduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths. Formulas for distributions of quantities such as degree sequence, shape, and total length are derived. An interpretation is given in terms of sampling from the inhomogeneous continuum random tree of Aldous and Pitman (1998). ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 176–195, 1999     

On the number of vertices of given degree in a random graph

This note can be treated as a supplement to a paper written by Bollobas which was devoted to the vertices of a given degree in a random graph. We determine some values of the edge probability p for which the number of vertices of a given degree of a random graph G ∈ ??(n, p) asymptotically has a normal distribution.     

New upper bounds on the spectral radius of trees with the given number of vertices and maximum degree

This paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree. We obtain the new upper bounds on the spectral radius of the trees, and the results are the best upper bounds expressed by the number of vertices and maximum degree, at present.     

The Estrada index of trees

The Estrada index of a graph G is defined as , where λ₁,λ₂,…,λ_n are the eigenvalues of its adjacency matrix. We determine the unique tree with maximum Estrada index among the set of trees with given number of pendant vertices. As applications, we determine trees with maximum Estrada index among the set of trees with given matching number, independence number, and domination number, respectively. Finally, we give a proof of a conjecture in [J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem. 64 (2010) 799-810] on trees with minimum Estrada index among the set of trees with two adjacent vertices of maximum degree.     

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 distance spectral radius of trees

The unique graphs with maximum distance spectral radius among trees with given number of vertices of maximum degree and among homeomorphically irreducible trees, respectively, are determined.     

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     

Expected numbers at hitting times

We determine exactly the expected number of hamilton cycles in the random graph obtained by starting with n isolated vertices and adding edges at random until each vertex degree is at least two. This complements recent work of Cooper and Frieze. There are similar results concerning expected numbers, for example, of perfect matchings, spanning trees, hamilton paths, and directed hamilton cycles.     

Competing first passage percolation on random graphs with finite variance degrees

We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate λ₁ (λ₂) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if λ₁ = λ₂, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V ∈ (0,1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If λ₁ ≠ λ₂, on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.     

An edge-swap heuristic for generating spanning trees with minimum number of branch vertices

This paper presents a new edge-swap heuristic for generating spanning trees with a minimum number of branch vertices, i.e. vertices of degree greater than two. This problem was introduced in Gargano et al. (Lect Notes Comput Sci 2380:355–365, 2002) and has been called the minimum branch vertices problem by Cerulli et al. (Comput Optim Appl 42:353–370, 2009). The heuristic starts with a random spanning tree and iteratively reduces the number of branch vertices by swapping tree edges with edges not currently in the tree. It can be easily implemented as a multi-start heuristic. We report on extensive computational experiments comparing single-start and multi-start variants on our heuristic with other heuristics previously proposed in the literature.     

Rayleigh processes, real trees, and root growth with re-grafting

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N→∞ of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N→∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–Hausdorff distance to metrize the space of rooted compact real trees. Berkeley Statistics Technical Report No. 654 (February 2004), revised October 2004. To appear in Probability Theory and Related Fields. SNE supported in part by NSF grants DMS-0071468 and DMS-0405778, and a Miller Institute for Basic Research in Science research professorship JP supported in part by NSF grants DMS-0071448 and DMS-0405779 AW supported by a DFG Forchungsstipendium     

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.