### Analysis of some statistics for increasing tree families

*Alois Panholzer, Helmut Prodinger*

#### 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*depth*of a randomly chosen node,*distance*between two randomly chosen nodes, and the generalisations where*p*nodes 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.Full Text: GZIP Compressed PostScript PostScript PDF original HTML abstract page