  In this article we focus on a general model of random walk on random marked
trees. We prove a recurrence criterion, analogue to the recurrence criterion
proved by R. Lyons and Robin Pemantle (1992) in a slightly different model. In
the critical case, we obtain a criterion for the positive/null recurrence.
Several regimes appear, as proved (in a similar model), by Y. Hu and Z. Shi
(2007). We focus on the "diffusive" regime and improve their result in this
case, by obtaining a functional Central Limit Theorem. Our result is also an
extension of a result by Y. Peres and O. Zeitouni (2008), obtained in the
setting of biased random walk in Galton-Watson trees.
