We study basic spectral features of graph Laplacians associated to a class of
rooted trees which contains all regular trees. Trees in this class can be
generated by substitution processes. Their spectra are shown to be purely
absolutely continuous and to consist of finitely many bands. The main result
gives stability of absolutely continuous spectrum under sufficiently small
radially label symmetric perturbations for non regular trees in this class. In
sharp contrast, the absolutely continuous spectrum can be completely destroyed
by arbitrary small radially label symmetric perturbations for regular trees in
this class.