  We prove that the size of the largest common subtree between two uniform,
independent, leaf-labelled random binary trees of size $n$ is typically less
than $n^{1/2-\varepsilon}$ for some $\varepsilon>0$. 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 $(1-\varepsilon)$-H\"older homeomorphism
between two independent copies of the Brownian tree.
