  Three standard subtree transfer operations for binary trees, used in
particular for phylogenetic trees, are: tree bisection and reconnection
($TBR$), subtree prune and regraft ($SPR$) and rooted subtree prune and regraft
($rSPR$). For a pair of leaf-labelled binary trees with $n$ leaves, the maximum
number of such moves required to transform one into the other is
$n-\Theta(\sqrt{n})$, extending a result of Ding, Grunewald and Humphries. We
show that if the pair is chosen uniformly at random, then the expected number
of moves required to transfer one into the other is $n-\Theta(n^{2/3})$. These
results may be phrased in terms of agreement forests: we also give extensions
for more than two binary trees.
