  We prove that every amenable one-ended Cayley graph has an invariant spanning
tree of one end. More generally, for any 1-ended amenable unimodular random
graph we construct a factor of iid percolation (jointly unimodular subgraph)
that is almost surely a spanning tree of one end. In [2] and [1] similar claims
were proved, but the resulting spanning tree had 1 or 2 ends, and one had no
control of which of these two options would be the case.
