  A BMW group of degree $(m,n)$ is a group that acts simply transitively on
vertices of the product of two regular trees of degrees $m$ and $n$. We show
that the number of commensurability classes of BMW groups of degree $(m,n)$ is
bounded between $(mn)^{\alpha mn}$ and $(mn)^{\beta mn}$ for some
$0<\alpha<\beta$. In fact, we show that the same bounds hold for virtually
simple BMW groups. We introduce a random model for BMW groups of degree $(m,n)$
and show that asymptotically almost surely a random BMW group in this model is
irreducible and hereditarily just-infinite.
