  The motivation for this paper is the study of the phase transition for
recurrence/transience of a class of self-interacting random walks on trees,
which includes the once-reinforced random walk. For this purpose, we define a
quantity, that we call the branching-ruin number of a tree, which provides (in
the spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measure
trees with polynomial growth. We prove that the branching-ruin number of a tree
is equal to the critical parameter for the recurrence/transience of the
once-reinforced random walk. We define a sharp and effective (i.e. computable)
criterion characterizing the recurrence/transience of a larger class of
self-interacting walks on trees, providing the complete picture for their phase
transition.
