  In this paper we consider a random walk in random environment on a tree and
focus on the boundary case for the underlying branching potential. We study the
range $R\_n$ of this walk up to time $n$ and obtain its correct asymptotic in
probability which is of order $n/\log n$. Thisresult is a consequence of the
asymptotical behavior of the number of visited sites at generations of order
$(\log n)^2$,which turn out to be the most visited generations. Our proof which
involves a quenched analysisgives a description of the typical environments
responsible for the behavior of $R\_n$.
