  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 $\mathrm{Sub}(G)$ 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.
