Let p(x, y) be the transition probability of an isotropic random walk on a tree, where each site has d [greater-or-equal, slanted]3 neighbors. We define a branching random walk by letting a particle at site x give birth to a new particle at site y at rate [lambda]dp(x, y), jump to y at rate vdp(x, y), and die at rate [delta]. Let [lambda]2 (respectively, [mu]2) be the infimum of [lambda] such that the process starting with one particle has positive probability of surviving forever (respectively, of having a fixed site occupied at arbitrarily large times). We compute [lambda]2 and [mu]2 exactly, proving that [lambda]2<[mu]2: i.e., the process has two phase transitions. We characterize [lambda]2 (respectively, [mu]2) in terms of the expected number of particles on the tree (respectively, at a fixed site). We also prove similar results for the biased voter model. Finally, for the contact process, branching random walk and biased voter model on the tree, we prove that the second phase transition has a discontinuity which is absent in Euclidian space
