The maximum agreement subtree problem

TL;DR

This paper studies the maximum size of common subtrees between two binary phylogenetic trees, establishing new lower bounds and structural results that improve understanding of their shared substructure.

Contribution

The paper proves a new lower bound of 1(\u221a{ log n}) for the largest common subtree, improving previous bounds, and introduces a Ramsey-type result for binary trees.

Findings

Any two binary phylogenetic trees have a common subtree on 1({ log n}) leaves.

When one tree is balanced or a caterpillar, the largest common subtree has 1( log n) leaves.

Balanced trees share a common subtree on a constant fraction of leaves, n^, for some  > 0.

Abstract

In this paper we investigate an extremal problem on binary phylogenetic trees. Given two such trees $T_1$ and $T_2$, both with leaf-set ${1,2,...,n}$, we are interested in the size of the largest subset $S \subseteq {1,2,...,n}$ of leaves in a common subtree of $T_1$ and $T_2$. We show that any two binary phylogenetic trees have a common subtree on $\Omega(\sqrt{\log{n}})$ leaves, thus improving on the previously known bound of $\Omega(\log\log n)$ due to M. Steel and L. Szekely. To achieve this improved bound, we first consider two special cases of the problem: when one of the trees is balanced or a caterpillar, we show that the largest common subtree has $\Omega(\log n)$ leaves. We then handle the general case by proving and applying a Ramsey-type result: that every binary tree contains either a large balanced subtree or a large caterpillar. We also show that there are constants $c, \alpha > 0$ such that, when both trees are balanced, they have a common subtree on $c n^\alpha$ leaves. We conjecture that it is possible to take $\alpha = 1/2$ in the unrooted case, and both $c = 1$ and $\alpha = 1/2$ in the rooted case.