Maximum agreement subtrees and Hölder homeomorphisms between Brownian trees

We prove that the size of the largest common subtree between two uniform, independent, leaf-labeled random binary trees of size n is typically less than n 1/2-ε for some ε>0. Our proof relies on the coupling between discrete random trees and the Brownian tree and on a recursive decomposition of the Brownian tree due to Aldous. Along the way, we also show that almost surely, there is no (1-ε)-Hölder homeomorphism between two independent copies of the Brownian tree.