Coalescence in small generations for the diffusive randomly biased walk on Galton-Watson trees

We investigate the range $\mathcal{R}_T$ of the diffusive biased walk $\mathbb{X}$ on a Galton-Watson tree $\mathbb{T}$ in random environment, that is to say the sub-tree of $\mathbb{T}$ of all distinct vertices visited by this walk up to the time $T$. We study the volume of the range with constraints and more precisely the number of $k$-tuples ($k\geq 2$) of distinct vertices in this sub-tree, in small generations and satisfying an hereditary condition. A special attention is paid to the vertices visited during distinct excursions of $\mathbb{X}$ above the root of the Galton-Watson tree as we observe they give the major contribution to this range. As an application, we study the genealogy of $k\geq 2$ distinct vertices of the tree $\mathcal{R}_T$ picked uniformly from those in small generations. It turns out that two or more vertices among them share a common ancestor for the last time in the remote past. We also point out an hereditary character in their genealogical tree due to the random environment.