On the Cantor-Bendixson rank of the Grigorchuk group and the Gupta-Sidki $3$ group

We study the Cantor–Bendixson rank of the space of subgroups for members of a general class of finitely generated self-replicating branch groups. In particular, we show for $G$ either the Grigorchuk group or the Gupta–Sidki $3$ group, the Cantor–Bendixson rank of $\mathrm{S}\mathrm{u}\mathrm{b}(G)$ is $\omega$. For each natural number $n$, we additionally characterize the subgroups of rank $n$ and give a description of subgroups in the perfect kernel.