# On the Extremal Maximum Agreement Subtree Problem

## TL;DR

This paper proves that the minimum size of the maximum agreement subtree for any two phylogenetic trees with n leaves grows asymptotically as Theta(log n), resolving a long-standing open problem.

## Contribution

It establishes the asymptotic growth rate of mast(n) as Theta(log n), closing the gap between previous bounds for the extremal agreement subtree problem.

## Key findings

- mast(n) grows asymptotically as Theta(log n)
- Resolved the long-standing open problem on agreement subtree size
- Provided tight bounds on the extremal agreement subtree problem

## Abstract

Given two phylogenetic trees with the $\{1, \ldots, n\}$ leaf-set the maximum agreement subtree problem asks what is the maximum size of the subset $A \subseteq \{1, \ldots, n\}$ such that the two trees are equivalent when restricted to $A$. The long-standing extremal version of this problem focuses on the smallest number of leaves, $\mathrm{mast}(n)$, on which any two (binary and unrooted) phylogenetic trees with $n$ leaves must agree. In this work we prove that this number grows asymptotically as $\Theta(\log n)$; thus closing the enduring gap between the lower and upper asymptotic bounds on $\mathrm{mast}(n)$.