On the Largest Common Subtree of Random Leaf-Labeled Binary Trees

TL;DR

This paper improves the lower bound on the size of the largest common subtree of two independent random binary trees, using recursive splitting and stochastic fragmentation methods, establishing a new bound of order n^0.366.

Contribution

The paper introduces a novel recursive splitting construction and analysis to improve the lower bound on the largest common subtree size.

Findings

Lower bound improved to n^0.366

Uses recursive splitting and stochastic fragmentation methods

Highlights the challenge of improving the upper bound

Abstract

The size of the largest common subtree (maximum agreement subtree) of two independent uniform random binary trees on $n$ leaves is known to be between orders $n^{1/8}$ and $n^{1/2}$. By a construction based on recursive splitting and analyzable by standard "stochastic fragmentation" methods, we improve the lower bound to order $n^\beta$ for $\beta = \frac{\sqrt{3} - 1}{2} = 0.366$. Improving the upper bound remains a challenging problem.