  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.
