It has hitherto been known that in a transitive unimodular graph, each tree in the wired spanning forest has only one end a.s. We dispense with the assumptions of transitivity and unimodularity, replacing them with a much broader condition on the isoperimetric profile that requires just slightly more than uniform transience.
Russell Lyons. Benjamin Morris. Oded Schramm. "Ends in Uniform Spanning Forests." Electron. J. Probab. 13 1702 - 1725, 2008. https://doi.org/10.1214/EJP.v13-566