In this work, we are interested in the set of visited vertices of a tree T by a randomly biased random walk X := (Xn,n ∈ N). The aim is to study a generalized range, that is to say the volume of the trace of X with both constraints on the trajectories of X and on the trajectories of the underlying branching random potential V := (V(x),x ∈ 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 X and the ones of V.