We discuss constructions and applications of definable one-ended spanning trees and, more strongly, connected toasts in one-ended Borel graphs. In particular, we provide Borel algorithms for rounding flows and finding balanced orientations in graphs that admit these structures and deduce a variant of Gardner's equidecomposition conjecture that applies to sets that are ``uniformly distributed" with respect to an action of a one-ended amenable group.

This is based on joint work with Kun and Sabok and Poulin and Zomback.