Branch groups, orbit growth, and subgroup growth types for pro-p groups

In their book Subgroup Growth, Lubotzky and Segal asked what are the possible types of subgroup growth of pro-$p$ group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta-Sidki groups which have all possible types of subgroup growth between $n^{(\log n)^2}$ and $e^n$. Thus, we give an almost complete answer to Lubotzky and Segal's question. In addition, we show that a class of pro-$p$ branch groups, including the Grigorchuk group and the Gupta-Sidki groups, all have subgroup growth type $n^{\log n}$.