In this paper we consider a random walk in random environment on a tree and focus on the frontier 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 the order of $n / \log n$. This result 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 analysis gives a description of the typical environments responsible for the behavior of $R_n$.