Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

Abstract

Limit theorems are established for some functionals of the distances between two nodes in weighted random b-ary recursive trees. We consider the depth of the nth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.

Building Random Trees From Blocks

“…(The recursive tree is a very special class of our model.) The depth of nodes in b-ary recursive trees (increasing trees with restricted outdegrees) is studied in [11], and the height of a generalized class of edge-weighted random trees is studied in [4]. This general class includes as special cases random binary search trees, random recursive trees, random plane oriented trees, and random split trees.…”

On the Asymptotic Internal Path Length and the Asymptotic Wiener Index of Random Split Trees

“…For the class of random increasing trees, which covers in particular the random recursive tree and the plane oriented recursive tree, the second order asymptotic of the expectation of the internal path length is derived in Bergeron et al (1992). In Munsonius and Rüschendorf (2010) the asymptotic behavior of the expectation and a limit theorem for the internal path length of random b-ary trees with weighted edges is proved. By special choices of the edge weights, the analogous results are obtained for the class of random linear recursive trees, which encompasses in particular the random plane oriented recursive tree.…”

On martingale tail sums for the path length in random trees

“…In the general case, it was established recently by Broutin and Holmgren [4] and independently by Munsonius [29]. Finally, for the family of weighted b-ary trees introduced by Broutin and Devroye [3], the path length was analyzed by Munsonius and Rüschendorf [30]. In recent years, deeper results on the profile of random trees, that is on the number of nodes on a given level (i.e.…”

“…< m ∈ N, and to[29, Theorem 5.6] for β ∈ N 0 , m = 1 from which the assertion can be deduced by checking the conditions formulated in the proof of Lemma 4.3 in…”