  We characterize the structural impediments to the existence of Borel perfect
matchings for acyclic locally countable Borel graphs admitting a Borel
selection of finitely many ends from their connected components. In particular,
this yields the existence of Borel matchings for such graphs of degree at least
three. As a corollary, it follows that acyclic locally countable Borel graphs
of degree at least three generating $\mu$-hyperfinite equivalence relations
admit $\mu$-measurable matchings. We establish the analogous result for Baire
measurable matchings in the locally finite case, and provide a counterexample
in the locally countable case.
