We will discuss the spectral properties of random operators on regular tree graphs. The models have have been among the earliest studied for Anderson localization, and they continue to attract attention because of analogies with localization issues for many particles. The talk will focus on the location of the mobility edge. Somewhat surprisingly, a resonance mechanism will be proven to cause the appearance of absolutely continuous spectrum in a regime extending well beyond the energy band of the operator's non-random hopping term. For weak disorder, this includes a Lifshitz tail regime of very low density of states.