We study actions of a countable group on a countable tree, and the orbit equivalence relation of the induced action on the Gromov boundary. We will identify a natural condition which implies that this equivalence relation is hyperfinite, and discuss some examples. We also identify a natural weakening of the aforementioned condition which implies measure hyperfiniteness of the boundary action. Finally, we document some examples of group actions on trees whose boundary action is not hyperfinite. This is based on joint work with Srivatsav Kunnawalkam Elayavalli, Koichi Oyakawa, and Forte Shinko.