  Let $(X,\tau)$ be a Polish space with Borel probability measure $\mu,$ and
$G$ a locally finite one-ended Borel graph on $X.$ We show that $G$ admits a
Borel one-ended spanning tree generically. If $G$ is induced by a free Borel
action of an amenable (resp., polynomial growth) group then we show the same
result $\mu$-a.e. (resp., everywhere). Our results generalize recent work of
Tim\'ar, as well as of Conley, Gaboriau, Marks, and Tucker-Drob, who proved
this in the probability measure preserving setting. We apply our theorem to
find Borel orientations in even degree graphs and measurable and Baire
measurable perfect matchings in regular bipartite graphs, refining theorems
that were previously only known to hold for measure preserving graphs. In
particular, we prove that bipartite one-ended $d$-regular Borel graphs admit
Baire measurable perfect matchings.
