Bounds on the expected size of the maximum agreement subtree for a given tree shape

TL;DR

This paper establishes that the expected size of the maximum agreement subtree between two random trees of the same shape scales with the square root of the number of leaves, using structural decomposition and probabilistic methods.

Contribution

It provides tight bounds on the expected maximum agreement subtree size for trees of a given shape, introducing a novel decomposition technique and a generalized probabilistic argument.

Findings

Expected maximum agreement subtree size is proportional to √n.

Structural decomposition into blobs aids in analysis.

Generalized probabilistic approach applies to exchangeable tree distributions.

Abstract

We show that the expected size of the maximum agreement subtree of two $n$-leaf trees, uniformly random among all trees with the shape, is $\Theta(\sqrt{n})$. To derive the lower bound, we prove a global structural result on a decomposition of rooted binary trees into subgroups of leaves called blobs. To obtain the upper bound, we generalize a first moment argument for random tree distributions that are exchangeable and not necessarily sampling consistent.