Analysis of some statistics for increasing tree families

Abstract

This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many mportant tree families that have applications in computer science or are used as probabilistic models in various applications, like recursive trees, heap-ordered trees or binary increasing trees (isomorphic to binary search trees) are members of this variety of trees. We consider the parameters depthof a randomly chosen node, distancebetween two randomly chosen nodes, and the generalisations where pnodes are randomly chosen Under the restriction that the node-degrees are bounded, we can prove that all these parameters converge in law to the Normal distribution. This extends results obtained earlier for binary search trees and heap-ordered trees to a much larger class of structures.