  We provide a unified proof of all known examples of locally compact groups
that enjoy the Howe-Moore property, namely, the vanishing at infinity of all
matrix coefficients of the group unitary representations that are without
non-zero invariant vectors. These examples are: connected, non-compact, simple
real Lie groups with finite center, isotropic simple algebraic groups over non
Archimedean local fields and closed, topologically simple subgroups of Aut(T)
that act 2-transitively on the boundary at infinity of T, where T is a
bi-regular tree with valence > 2 at every vertex.
