  We are interested in the biased random walk on a supercritical Galton--Watson
tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow
movement. In order to observe such a slow movement, the bias needs to be
random; the resulting random walk is then a tree-valued random walk in random
environment. We investigate the recurrent case, and prove, under suitable
general integrability assumptions, that upon the system's non-extinction, the
maximal displacement of the walk in the first n steps, divided by (log n)^3,
converges almost surely to a known positive constant.
