On tail bounds for random recursive trees

TL;DR

This paper derives upper and lower tail bounds for key parameters of random recursive trees, such as internal path length and Wiener index, using Chernoff bounds and stochastic domination techniques.

Contribution

It introduces new tail bounds for multivariate recursive distributions in random trees, extending previous analyses to weighted and b-ary recursive trees.

Findings

Upper tail bounds for internal path length and Wiener index

Lower tail bounds for Wiener index in recursive trees

Application of Chernoff bounds and stochastic domination techniques

Abstract

We consider a multivariate distributional recursion of sum-type as arising in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.