  In this work, we are interested in the set of visited vertices of a tree
$\mathbb{T}$ by a randomly biased random walk $\mathbb{X}:=(X_n,n \in
\mathbb{N})$. The aim is to study a generalized range, that is to say the
volume of the trace of $\mathbb{X}$ with both constraints on the trajectories
of $\mathbb{X}$ and on the trajectories of the underlying branching random
potential $\mathbb{V}:=(V(x), x \in \mathbb{T})$. Focusing on slow regime's
random walks (see [HS16b], [AC18]), we prove a general result and detail
examples. These examples exhibit many different behaviors for a wide variety of
ranges, showing the interactions between the trajectories of $\mathbb{X}$ and
the ones of $\mathbb{V}$.
