  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.
