Average Case Analysis of Leaf-Centric Binary Tree Sources

TL;DR

This work studies the average number of distinct fringe subtrees in random binary trees generated by leaf-centric binary tree sources, unifying and extending classic results for binary search trees and uniformly random binary trees. By defining $F_{n,\sigma}$ and leveraging a cut-point analysis, the authors derive upper and lower bounds on $\mathbb{E}[F_{n,\sigma}]$ across several source classes: $\psi$-upper-bounded, $\phi$-weakly-balanced, and $\vartheta$-strongly-balanced sources, as well as $\xi$-unbalanced sources. Key findings include $\mathbb{E}[F_{n,\sigma}] = O\big(n\psi(\log_4 n)\big)$ for appropriate $\psi$, and tight lower bounds of $\mathbb{E}[F_{n,\sigma}] = \Omega\big(n/\log n\big)$ or $\Theta\big(n/\sqrt{\log n}\big)$ in several regimes, with concrete confirmation for the BST, binomial, and uniform models; the uniform model yields $\Theta\big(n/\sqrt{\log n}\big)$, matching classical results. The work also includes open problems, notably the exact asymptotics for the critical $\beta$-splitting model and extensions to other tree-source paradigms, underscoring the versatility of leaf-centric sources for fringe-subtree analyses.

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 $Θ(n/\log n)$, 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 $Θ(n/\sqrt{\log n})$.

Figures (3)

• Figure 1: A binary tree (left) and its corresponding minimal DAG (right).
• Figure 2: Two binary trees $t_1$ (left) and $t_2$ (right), with the probabilities assigned to them by the binary search tree model (Example \ref{['ex:bst']}), the uniform model (Example \ref{['ex:uniform']}) and the binomial random tree model (Example \ref{['ex:dst']}) with $p=1/4$.
• Figure 3: A binary tree (left) and its six distinct fringe subtrees (right).

Theorems & Definitions (52)

• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Lemma 2.7
• proof
• Definition 3.1: $\psi$-upper-bounded sources