Width and mode of the profile for some random trees of logarithmic height

Abstract

We propose a new, direct, correlation-free approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width and almost sure convergence. It is widely applicable to random trees of logarithmic height, including recursive trees, binary search trees, quad trees, plane-oriented ordered trees and other varieties of increasing trees. 1. Introduction. Most random trees in the discrete probability literature have height either of order √ n or of order log n (n being the tree size); see [1]. For simplicity, we call these trees square-root trees and log trees, respectively. Profiles (number of nodes at each level of the tree) of random square-root trees have a rich connection to diverse structures in combinatorics and in probability, and have been extensively studied. In contrast, profiles of random log trees, arising mostly from data structures and computer algorithms, were less addressed and only quite recently were their limit distributions, drastically different from those of square-root trees, better understood; see [3, 12, 13, 21, 27] and the references therein. We study in this paper the asymptotics of width, which is defined to be the size of the most abundant level, and its close connection to the profile. There are many results on first-order asymptotics of profiles for standard log trees, such as binary search trees, random recursive trees, m-ary search trees and quad trees. In some cases, quite accurate asymptotic expressions are known for the expected profile. There is already a paucity of results with regard to