Hosam M. Mahmoud

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 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.

Analysis of quickselect : an algorithm for order statistics

We study QUICKSELECT a one-sided version of QUICKSORT suited for finding the order statistics of a sample. We identify procedures by which the moments of the number of comparisons can be found exactly under both assumptions that the order statistic in question is randomly chosen or fixed. The procedure is illustrated by finding the exact mean and variance for a randomly selected order statistic as well as the first few in the fixed case. The existence of an absolutely continuous infinitely divisible limit law with asymmetric left and right tails is demonstrated in the case of a randomly chosen order statistic. Some of these distributional properties carry over to the case of a very small fixed order statistic.

On generalized Polya urn models

We consider a general two-color urn model characterized by a 2x2 matrix of integerswithout constraints on the values of these four integers other than non-negativity. Exact distributions of the number of balls of a specific color that appear after n draws are presented. Via Poissonization, we obtain an asymptotic mean for a specific example to illustrate that asymptotics of bona fide nature may appear in non-classical urn models.

Asymptotic Joint Normality of Outdegrees of Nodes in Random Recursive Trees

We study the joint probability distribution of the number of nodes of outdegree 0, 1, and 2 in a random recursive tree. We complete the known partial list of exact means and variances for outdegrees up to two by obtaining exact combinatorial expressions for the remaining means, variances, and covariances. The joint probability distribution of the number of nodes of outdegree 0, 1, and 2 is shown to be asymptotically trivariate normal and the asymptotic covariance structure is explicitly determined. It is also shown how to extend the results (at least in principle) to obtain a limiting multivariate normal distribution for nodes of outdegree 0, 1, …, k.

Distances in random plane-oriented recursive trees

Abstract The average number of nodes in a stratum of random plane-oriented recursive trees is found. The expression is used to determine the exact probability distribution of the depth of thenth node. It is further shown that the limiting distribution of the normalized depth of this node is the standard normal distribution. Via martingales, the normalized external path length is shown to converge almost surely and in L2 to a limiting random variable.

The Joint Distribution of Elastic Buckets in Multiway Search Trees

Random search trees are studied when they grow under a general computer memory management scheme. In a general scheme, the space is released in buckets of certain predesignated sizes. For a search tree with branch factor

Average-case analysis of multiple Quickselect: An algorithm for finding order statistics

m

On the average internal path length of many search trees

, the nodes may hold up to

Analytic variations on bucket selection and sorting

m-1

On rotations in fringe-balanced binary trees

keys. Suppose the buckets of the memory management scheme that can hold less than