We prove amenability for a broad class of equivalence relations which have trees associated to the equivalence classes. The proof makes crucial use of percolation on trees. We also discuss related concepts and results, including amenability of automorphism groups. A second main result is that no discrete subgroup of the automorphism group of a tree is isomorphic to the fundamental group of any closed manifold M admitting a nontrivial connection-preserving, volume-preserving action of a noncompact, simply connected, almost simple Lie group having Kazhdan's property (T). The technique of proof also shows that M does not admit a hyperbolic structure.