  In this paper we are interested in a random walk in a random environment on a
super-critical Galton-Watson tree. We focus on the recurrent cases already
studied by Y. Hu and Z. Shi and G. Faraud. We prove that the largest generation
entirely visited by these walks behaves like log n and that the constant of
normalization which differs from a case to another is function of the inverse
of the constant of Biggins' law of large number for branching random walks.
