We are interested in the random walk in random environment (RWRE) on an infinite tree. Lyons and Pemantle (1992) gave a precise recurrence/transience criterion. We shall focus on the almost sure asymptotic behaviors of a recurrent RWRE on tree: under some general assumptions upon the distribution of the environment, the walk may be subdiffusive or may exhibit an unusual slow movement, which leads us to study some fine path properties on the branching random walk.