Expected number of induced subtrees shared by two independent copies of a random tree

TL;DR

This paper analyzes the expected size of the largest common induced subtree shared by two independent random trees, extending known bounds from binary trees to a broader class of Galton–Watson trees with mean one offspring.

Contribution

It generalizes the known $O(n^{1/2})$ bound for maximum agreement subtree size from binary trees to a wider class of Galton–Watson trees conditioned on leaf count.

Findings

The likely magnitude of MAST is $O(n^{1/2})$ for the considered class of trees.

The bound holds for trees generated by a Galton–Watson process with mean 1 offspring.

The result extends previous bounds from binary trees to more general random tree models.

Abstract

Consider a rooted tree $T$ with leaf-set $[n]$, and with all non-leaf vertices having out-degree $2$, at least. A rooted tree $\mathcal T$ with leaf-set $S\subset [n]$ is induced by $S$ in $T$ if $\mathcal T$ is the lowest common ancestor subtree for $S$, with all its degree-2 vertices suppressed. A "maximum agreement subtree" (MAST) for a pair of two trees $T'$ and $T"$ is a tree $\mathcal T$ with a largest leaf-set $S\subset [n]$ such that $\mathcal T$ is induced by $S$ both in $T'$ and $T"$. Bryant et al. \cite{BryMcKSte} and Bernstein et al. \cite{Ber} proved, among other results, that for $T'$ and $T"$ being two independent copies of a random binary (uniform or Yule-Harding distributed) tree $T$, the likely magnitude order of $\text{MAST}(T',T")$ is $O(n^{1/2})$. We prove this bound for a wide class of random rooted trees : $T$ is a terminal tree of a branching, Galton--Watson, process with an ordered-offspring distribution of mean $1$, conditioned on "total number of leaves is $n$".