  We are interested in the random walk in random environment on an infinite
tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion.
Our paper focuses on the almost sure asymptotic behaviours of a recurrent
random walk $(X\_n)$ in random environment on a regular tree, which is closely
related to Mandelbrot [13]'s multiplicative cascade. We prove, under some
general assumptions upon the distribution of the environment, the existence of
a new exponent $\nu\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$
behaves asymptotically like $n^{\nu}$. The value of $\nu$ is explicitly
formulated in terms of the distribution of the environment.
