Average Case Analysis of Leaf-Centric Binary Tree Sources

TL;DR

This paper analyzes the average number of distinct fringe subtrees in random leaf-centric binary trees, generalizing previous results for binary search trees and uniform binary trees, revealing asymptotic behaviors.

Contribution

It extends the analysis of fringe subtrees to leaf-centric binary tree sources, providing generalized asymptotic results beyond prior specific models.

Findings

Average number of distinct fringe subtrees in leaf-centric trees is characterized.

Generalizes known asymptotic results for binary search and uniform trees.

Provides new bounds for diverse leaf-centric binary tree models.

Abstract

We study the average number of distinct fringe subtrees in random trees generated by leaf-centric binary tree sources as introduced by Zhang, Yang and Kieffer. A leaf-centric binary tree source induces for every $n \geq 2$ a probability distribution on the set of binary trees with $n$ leaves. We generalize a result by Flajolet, Gourdon, Martinez and Devroye, according to which the average number of distinct fringe subtrees in a random binary search tree of size $n$ is in $\Theta(n/\log n)$, as well as a result by Flajolet, Sipala and Steayert, according to which the number of distinct fringe subtrees in a uniformly random binary tree of size $n$ is in $\Theta(n/\sqrt{\log n})$.