  We study a class of rooted trees with a substitution type structure. These
trees are not necessarily regular, but exhibit a lot of symmetries. We consider
nearest neighbor operators which reflect the symmetries of the trees. The
spectrum of such operators is proven to be purely absolutely continuous and to
consist of finitely many intervals. We further investigate stability of the
absolutely continuous spectrum under perturbations by sufficiently small
potentials. On the one hand, we look at a class of deterministic potentials
which include radial symmetric ones. The absolutely continuous spectrum is
stable under sufficiently small perturbations of this type if and only if the
tree is not regular. On the other hand, we study random potentials. In this
case, we prove stability of absolutely continuous spectrum for both regular and
non regular trees provided the potentials are sufficiently small.
