The largest common subtree of two random trees

TL;DR

This paper analyzes the size and structure of the largest common subtree between two random Bienaymé trees, revealing that it typically scales with the square root of the tree size and has a specific path-based structure.

Contribution

It provides the first detailed asymptotic characterization of the largest common subtree in critical random trees, including tight bounds and structural insights.

Findings

LCS size is of order √n for critical trees with finite moments

LCS structure approximates three paths meeting at a central node

Largest common subtree can be significantly larger than √n in some cases

Abstract

We study the size and structure of the largest common subtree (LCS) between two independent Bienaym\'e trees conditioned to have size $n$. When the trees are critical with finite $2$nd and $(2+\kappa)$th moment respectively for some $\kappa>0$, we prove that the LCS has size of order $\sqrt{n}$, and is approximated by the length of three paths meeting at a central node. Moreover, we show that the largest common subtree between two critical independent Bienaym\'e trees with size $n$ and finite second moments may be much larger than $\sqrt{n}$, implying that our result is tight. We also pose a number of open questions and suggestions for future research.