  We study the spectra of quantum trees of finite cone type. These are quantum
graphs whose geometry has a certain homogeneity, and which carry a finite set
of edge lengths, coupling constants and potentials on the edges. We show the
spectrum consists of bands of purely absolutely continuous spectrum, along with
a discrete set of eigenvalues. Afterwards, we study random perturbations of
such trees, at the level of edge length and coupling, and prove the stability
of pure AC spectrum, along with resolvent estimates.
