For the random walk in random environment (RWRE) on Z, there are two well-known regimes: Sinai's slow movement in the recurrent case and Kesten-Kozlov-Spitzer's polynominal rate in the transient case. We discuss here a class of RWREs on trees and show that in the recurrent case, the walk may move slowly or may be subdiffusive according to the shape of some generating function. The asymptotic behavior of the RWRE on trees is closely related to that of some branching random walks.