  We give a counterexample to the conjecture of Martin and Thatte that two
balanced rooted binary leaf-labelled trees on $n$ leaves have a maximum
agreement subtree (MAST) of size at least $n^{\frac{1}{2}}$. In particular, we
show that for any $c>0$, there exist two balanced rooted binary leaf-labelled
trees on $n$ leaves such that any MAST for these two trees has size less than
$c n^{\frac{1}{2}}$. We also improve the lower bound of the size of such a MAST
to $n^{\frac{1}{6}}$.
