We investigate invariant random subgroups in groups acting on rooted trees. Let Alt(f)
(T) be the group of finitary even automorphisms of the d-ary rooted tree T. We prove
that a nontrivial ergodic invariant random subgroup (IRS) of Alt(f) (T) that acts
without fixed points on the boundary of T contains a level stabilizer, in particular
it is the random conjugate of a finite index subgroup.Applying the technique to branch
groups we prove that an ergodic IRS in a finitary regular branch group contains the
derived subgroup of a generalized rigid level stabilizer. We also prove that every
weakly branch group has continuum many distinct atomless ergodic IRS's. This extends
a result of Benli, Grigorchuk and Nagnibeda who exhibit a group of intermediate growth
with this property.
