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

Abstract

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 Polya 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. 0 1993 John Wiley & Sons, Inc.

The Diameter of a Scale-Free Random Graph

“…The most studied case is that of a uniform choice, but probabilities essentially proportional to degrees have also been considered. Such objects (where for the first vertex degree plus one is used) are known as random plane-oriented recursive trees, see [27,21], for example. Pittel [24] showed that the height (maximum distance of a vertex from the root) of such an object is (c + o(1)) log n with probability 1 − o (1), where c = (2γ) −1 for γ the solution of γe 1+γ = 1.…”

On the covariances of outdegrees in random plane recursive trees

“…Hence, the combined contribution of (4) and (5) is 3 − 24 (i + j + 4) 4 , which completes the proof.…”

“…In 2005 Janson [3], extending the earlier work of Mahmoud et al [4], established the joint asymptotic normality of the outdegrees of a random plane recursive tree (we refer to [3] for references, discussion, and statements, and to [2] for a much wider context). In particular, he gave the following formula for the entries of the limiting covariance matrix [ …”

Combinatorial Models for Cooperation Networks

“…where p n (d) denotes the probability that a random node in random PORT of size n has degree d (see [9] or [8]). Now, we introduce a substitution process that creates random graphs that have a global tree structure that is governed by plane oriented recursive trees.…”