  We study Random Walks in an i.i.d. Random Environment (RWRE) defined on
$b$-regular trees. We prove a functional central limit theorem (FCLT) for
transient processes, under a moment condition on the environment. We emphasize
that we make no uniform ellipticity assumptions. Our approach relies on
regenerative levels, i.e. levels that are visited exactly once. On the way, we
prove that the distance between consecutive regenerative levels have a
geometrically decaying tail. In the second part of this paper, we apply our
results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when the
process is defined on $b$-regular trees, with $ b \ge 4$, substantially
improving the results of the first author (see Theorem 3 of Collevecchio
(2006)).
