  We prove that in both the free and the wired uniform spanning forest (FUSF
and WUSF) of any unimodular random rooted network (in particular, of any Cayley
graph), it is impossible to distinguish the connected components of the forest
from each other by invariantly defined graph properties almost surely. This
confirms a conjecture of Benjamini, Lyons, Peres and Schramm.
  We use this to answer positively two additional questions of Benjamini,
Lyons, Peres and Schramm under the assumption of unimodularity. We prove that
on any unimodular random rooted network, the FUSF is either connected or has
infinitely many connected components almost surely, and, if the FUSF and WUSF
are distinct, then every component of the FUSF is transient and
infinitely-ended almost surely. All of these results are new even for Cayley
graphs.
