  Consider a rooted tree $T$ with leaf-set $[n]$, and with all non-leaf
vertices having out-degree $2$, at least. A rooted tree $\mathcal T$ with
leaf-set $S\subset [n]$ is induced by $S$ in $T$ if $\mathcal T$ is the lowest
common ancestor subtree for $S$, with all its degree-2 vertices suppressed. A
"maximum agreement subtree" (MAST) for a pair of two trees $T'$ and $T"$ is a
tree $\mathcal T$ with a largest leaf-set $S\subset [n]$ such that $\mathcal T$
is induced by $S$ both in $T'$ and $T"$. Bryant et al. \cite{BryMcKSte} and
Bernstein et al. \cite{Ber} proved, among other results, that for $T'$ and $T"$
being two independent copies of a random binary (uniform or Yule-Harding
distributed) tree $T$, the likely magnitude order of $\text{MAST}(T',T")$ is
$O(n^{1/2})$. We prove this bound for a wide class of random rooted trees : $T$
is a terminal tree of a branching, Galton--Watson, process with an
ordered-offspring distribution of mean $1$, conditioned on "total number of
leaves is $n$".
