Search trees: Metric aspects and strong limit theorems

TL;DR

This paper studies the metric properties of random binary search trees generated by standard algorithms, proving their convergence and deriving limit theorems for various tree functionals.

Contribution

It introduces a subtree size metric on search trees and establishes almost sure convergence of the associated metric spaces and tree functionals.

Findings

Metric spaces of search trees converge almost surely

Limit distributions of tree functionals are characterized as functions of the limit tree

Provides a framework for analyzing asymptotic properties of search trees

Abstract

We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces converge with probability 1. This is then used to obtain almost sure convergence for various tree functionals, together with representations of the respective limit random variables as functions of the limit tree.