Trees and amenable equivalence relations

Let R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a “treeing“of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.