  We consider groups that act on spherically symmetric rooted trees and study
the associated representation of the group on the space of locally constant
functions on the boundary of the tree. We introduce and discuss the new notion
of locally 2-transitive actions. Assuming local 2-transitivity our main theorem
yields a precise decomposition of the boundary representation into irreducible
constituents.
  The method can be used to study Gelfand pairs and enables us to answer a
question of Grigorchuk. To provide examples, we analyse in detail the local
2-transitivity of GGS-groups. Moreover, our results can be used to determine
explicit formulae for zeta functions of induced representations defined by
Klopsch and the author.
