We establish a dichotomy theorem characterizing the circumstances under which
a treeable Borel equivalence relation E is essentially countable. Under
additional topological assumptions on the treeing, we in fact show that E is
essentially countable if and only if there is no continuous embedding of E1
into E. Our techniques also yield the first classical proof of the analogous
result for hypersmooth equivalence relations, and allow us to show that up to
continuous Kakutani embeddability, there is a minimum Borel function which is
not essentially countable-to-one