  We investigate analogues of some of the classical results in homogeneous
dynamics in non-linear setting. Let $G$ be a closed subgroup of the group of
automorphisms of a biregular tree and $\Gamma<G$ a discrete subgroup. For a
large class of groups $G$ we give a classification of probability measures on
$G/\Gamma$ invariant under horospherical subgroups. When $\Gamma$ is a
cocompact lattice, we prove unique ergodicity of the horospherical action. We
prove Hedlund's theorem for geometrically finite quotients. Finally, we study
equidistribution of large compact orbits.
