Let $G$ be a group acting on a locally finite tree $T$. Then the boundary $\partial T$ is a Cantor set on which $G$ acts by homeomorphisms (indeed by quasi-isometries under a suitable metric). However, even if $G$ is the full automorphism group of $T$, we can't get the full quasi-isometry group of $\partial T$, as the tree structure puts further restrictions on the action.

What ways are known of identifying those group actions by homeomorphisms on the Cantor set which arise as actions on the boundary of a locally finite tree? Is there a nice algebraic description of $\mathrm{Aut}(T)$ as a subgroup of $\mathrm{Homeo}(\partial T)$?