Commensurability growth of branch groups

Fixing a subgroup $\mathrm{\Gamma }$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\mathrm{\Delta }$ of $G$ with $[\mathrm{\Gamma }:\mathrm{\Gamma }\cap \mathrm{\Delta }][\mathrm{\Delta }:\mathrm{\Gamma }\cap \mathrm{\Delta }]=n$. For pairs $\mathrm{\Gamma }\le A$, where $A$ is the automorphism group of a $p$-regular tree and $\mathrm{\Gamma }$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $\mathrm{\Gamma }$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely ${\mathrm{\aleph }}_0$ for any $n=p^k$.