  We solve two well-known open problems on the Hausdorff dimension of branch
groups. Firstly, we completely determine the self-similar Hausdorff spectrum of
the group of $q$-adic automorphisms where $q$ is a prime power, answering a
question of Grigorchuk. Indeed, we take a further step and completely determine
its Hausdorff spectra restricted to the most important subclasses of
self-similar groups, providing examples differing drastically from the
previously known ones in the literature. Our proof relies on a new explicit
formula for the computation of the Hausdorff dimension of closed self-similar
groups and a generalization of iterated permutational wreath products.
  Secondly, we provide the first examples of just infinite branch pro-$p$
groups with trivial Hausdorff dimension, disproving a well-known conjecture of
Boston which has been open for over two decades.
