  We show that the transience or recurrence of a random walk in certain random
environments on an arbitrary infinite locally finite tree is determined by the
branching number of the tree, which is a measure of the average number of
branches per vertex. This generalizes and unifies previous work of the authors.
It also shows that the point of phase transition for edge-reinforced random
walk is likewise determined by the branching number of the tree. Finally, we
show that the branching number determines the rate of first-passage percolation
on trees, also known as the first-birth problem. Our techniques depend on
quasi-Bernoulli percolation and large deviation results.
