Commensurability growth of branch groups

Fixing a subgroup Γ in a group G, the commensurability growth function assigns to each n the cardinality of the set of subgroups [increment] of G with [Γ: Γ ∩ [increment]][[increment]: Γ ∩ [increment]] = n. For pairs Γ ≤ A, where A is the automorphism group of a p-regular rooted tree and Γ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups Γ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on p-regular rooted trees, this function is precisely ℵ₀ for any n = p^k.