  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.
