  Fixing a subgroup $\Gamma$ in a group $G$, the commensurability growth
function assigns to each $n$ the cardinality of the set of subgroups $\Delta$
of $G$ with $[\Gamma: \Gamma \cap \Delta][\Delta : \Gamma \cap \Delta] = n$.
For pairs $\Gamma \leq A$, where $A$ is the automorphism group of a $p$-regular
tree and $\Gamma$ is finitely generated, we show that this function can take on
finite, countable, or uncountable cardinals. For almost all known branch groups
$\Gamma$ (the first Grigorchuk group, the twisted twin Grigorchuk group,
Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this
function is precisely $\aleph_0$ for any $n = p^k$.
