An Improved Lower Bound 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 random leaf-labeled binary trees from approximately n^{0.366} to n^{0.4464} by a recursive construction method.

Contribution

It introduces a new recursive construction that enhances the known lower bound for the largest common subtree in random binary trees.

Findings

Lower bound improved to n^{0.4464}

Recursive construction based on centroid splitting

Method extends previous algorithms by Aldous

Abstract

It is known that the size of the largest common subtree (i.e., the maximum agreement subtree) of two independent random binary trees with $n$ given labeled leaves is of order between $n^{0.366}$ and $n^{1/2}$. We improve the lower bound to order $n^{0.4464}$ by constructing a common subtree recursively and by proving a lower bound for its asymptotic growth. The construction is a modification of an algorithm proposed by D. Aldous by splitting the tree at the centroid and by proceeding recursively.