  We give a descriptive construction of trees for multi-ended graphs, which
yields yet another proof of Stallings' theorem on ends of groups. Even though
our proof is, in principle, not very different from already existing proofs and
it draws ideas from a paper of Kr\"{o}n, it is written in a way that easily
adapts to the setting of countable Borel equivalence relations, leading to a
free decomposition result and a sufficient condition for treeability.
