The height of depth‐weighted random recursive trees

In this paper, we introduce a model of depth‐weighted random recursive trees, created by recursively joining a new leaf to an existing vertex . In this model, the probability of choosing depends on its depth in the tree. In particular, we assume that there is a function such that if has depth then its probability of being chosen is proportional to . We consider the expected value of the diameter of this model as determined by , and for various increasing we find expectations that range from polylogarithmic to linear.     

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     

Percolation on random recursive trees

We study Bernoulli bond percolation on a random recursive tree of size n with percolation parameter p(n) converging to 1 as n tends to infinity. The sizes of the percolation clusters are naturally stored in a tree structure. We prove convergence in distribution of this tree‐indexed process of cluster sizes to the genealogical tree of a continuous‐state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin [5]. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuous‐time destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 655–680, 2016     

On the covariance of the level sizes in random recursive trees

In this paper we study the covariance structure of the number of nodes k and l steps away from the root in random recursive trees. We give an analytic expression valid for all k, l and tree sizes N. The fraction of nodes k steps away from the root is a random probability distribution in k. The expression for the covariances allows us to show that the total variation distance between this (random) probability distribution and its mean converges in probability to zero. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 519–539, 2002     

Justifying the small‐world phenomenon via random recursive trees

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the technique is three‐fold: it is quite simple and provides short proofs, it is applicable to a broad variety of models including those incorporating preferential attachment, and it provides bounds with small constants. We illustrate this by proving, for the first time, logarithmic upper bounds for the diameters of the following well known models: the forest fire model, the copying model, the PageRank‐based selection model, the Aiello‐Chung‐Lu models, the generalized linear preference model, directed scale‐free graphs, the Cooper‐Frieze model, and random unordered increasing k‐trees. Our results shed light on why the small‐world phenomenon is observed in so many real‐world graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 201–224, 2017     

On the structure of random plane-oriented recursive trees and their branches

This paper is an investigation of the structural properties of random plane-oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Pólya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second-order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal. © 1993 John Wiley & Sons, Inc.     

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.     

Random covering of an interval and a variation of Kingman's coalescent

We consider the covering of [0, 1] by a large number of small random intervals. We show that a simple variation of Kingman's coalescent describes the emergence of macroscopic connected components. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

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     

On goodness-of-fit tests for multiple recursive random number generators

This paper considers the problem of reducing the computational time in testing uniformity for a full period multiple recursive generator (MRG). If a sequence of random numbers generated by a MRG is divided into even number of segments, say 2s, then the multinomial goodness-of-fit tests and the empirical distribution function goodness-of-fit tests calculated from the ith segment are the same as those of the (s + i)th segment. The equivalence properties of the goodness-of-fit test statistics for a MRG and its associated reverse and additive inverse MRGs are also discussed.     

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     

Counting trees with random walks

We give a simple proof of Tutte’s matrix-tree theorem, a well-known result providing a closed-form expression for the number of rooted spanning trees in a directed graph. Our proof stems from placing a random walk on a directed graph and then applying the Markov chain tree theorem to count trees. The connection between the two theorems is not new, but it appears that only one direction of the formal equivalence between them is readily available in the literature. The proof we now provide establishes the other direction. More generally, our approach is another example showing that random walks can serve as a powerful glue between graph theory and Markov chain theory, allowing formal statements from one side to be carried over to the other.     

Vertices of degree k in random unlabeled trees

Let ??_n be the class of unlabeled trees with n vertices, and denote by H _n a tree that is drawn uniformly at random from this set. The asymptotic behavior of the random variable deg_k(H _n) that counts vertices of degree k in H _n was studied, among others, by Drmota and Gittenberger in [J Graph Theory 31(3) (1999), 227–253], who showed that this quantity satisfies a central limit theorem. This result provides a very precise characterization of the “central region” of the distribution, but does not give any non‐trivial information about its tails. In this work, we study further the number of vertices of degree k in H _n. In particular, for k = ??((logn/(loglogn))^1/2) we show exponential‐type bounds for the probability that deg_k(H _n) deviates from its expectation. On the technical side, our proofs are based on the analysis of a randomized algorithm that generates unlabeled trees in the so‐called Boltzmann model. The analysis of such algorithms is quite well‐understood for classes of labeled graphs, see e.g. the work [Bernasconi et al., SODA '08: Proceedings of the 19th Annual ACM‐SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008, pp. 132–141; Bernasconi et al., Proceedings of the 11th International Workshop, APPROX 2008, and 12th International Workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization, Springer, Berlin, 2008, pp. 303–316] by Bernasconi, the first author, and Steger. Comparable algorithms for unlabeled classes are unfortunately much more complex. We demonstrate in this work that they can be analyzed very precisely for classes of unlabeled graphs as well. © 2011 Wiley Periodicals, Inc. J Graph Theory. 69:114‐130, 2012     

Uniform recursive trees: Branching structure and simple random downward walk

As models for spread of epidemics, family trees, etc., various authors have used a random tree called the uniform recursive tree. Its branching structure and the length of simple random downward walk (SRDW) on it are investigated in this paper. On the uniform recursive tree of size n, we first give the distribution law of ζn,m, the number of m-branches, whose asymptotic distribution is the Poisson distribution with parameter . We also give the joint distribution of the numbers of various branches and their covariance matrix. On Ln, the walk length of SRDW, we first give the exact expression of P(Ln=2). Finally, the asymptotic behavior of Ln is given.     

Sizes of the largest clusters for supercritical percolation on random recursive trees

We consider Bernoulli bond‐percolation on a random recursive tree of size , with supercritical parameter for some fixed. We show that with high probability, the largest cluster has size close to whereas the next largest clusters have size of order only and are distributed according to some Poisson random measure. Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 29–44, 2014     

Geometry of the vacant set left by random walk on random graphs,Wright's constants,and critical random graphs with prescribed degrees

We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by ?erný and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.     

On the shape of the fringe of various types of random trees

We analyze a fringe tree parameter w in a variety of settings, utilizing a variety of methods from the analysis of algorithms and data structures. Given a tree t and one of its leaves a, the w(t, a) parameter denotes the number of internal nodes in the subtree rooted at a's father. The closely related w?(t, a) parameter denotes the number of leaves, excluding a, in the subtree rooted at a's father. We define the cumulative w parameter as W(t) = Σ_aw(t, a), i.e. as the sum of w(t, a) over all leaves a of t. The w parameter not only plays an important rôle in the analysis of the Lempel–Ziv '77 data compression algorithm, but it is captivating from a combinatorial viewpoint too. In this report, we determine the asymptotic behavior of the w and W parameters on a variety of types of trees. In particular, we analyze simply generated trees, recursive trees, binary search trees, digital search trees, tries and Patricia tries. The final section of this report briefly summarizes and improves the previously known results about the w? parameter's behavior on tries and suffix trees, originally published in one author's thesis (see Analysis of the multiplicity matching parameter in suffix trees. Ph.D. Thesis, Purdue University, West Lafayette, IN, U.S.A., May 2005; Discrete Math. Theoret. Comput. Sci. 2005; AD :307–322; IEEE Trans. Inform. Theory 2007; 53 :1799–1813). This survey of new results about the w parameter is very instructive since a variety of different combinatorial methods are used in tandem to carry out the analysis. Copyright © 2008 John Wiley & Sons, Ltd.     

A distributional study of the path edge-covering numbers for random trees

We study for various tree families the distribution of the number of edge-disjoint paths required to cover the edges of a random tree of size n. For all tree families considered we can show a central limit theorem with expectation ∼μn and variance ∼νn with constants μ, ν depending on the specific tree family.     

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.