We consider unimodular random rooted trees (URTs) and invariant forests in
Cayley graphs. We show that URTs of bounded degree are the same as the law of
the component of the root in an invariant percolation on a regular tree. We use
this to give a new proof that URTs are sofic, a result of Elek. We show that
ends of invariant forests in the hyperbolic plane converge to ideal boundary
points. We also prove that uniform integrability of the degree distribution of
a family of finite graphs implies tightness of that family for local
convergence, also known as random weak convergence.Comment: 19 pages, 4 figure
