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We investigate the range $\mathcal{R}_T$ of the diffusive biased walk$\mathbb{X}$ on a Galton-Watson tree $\mathbb{T}$ in random environment, thatis to say the sub-tree of $\mathbb{T}$ of all distinct vertices visited by thiswalk up to the time $T$. We study the volume of the range with constraints andmore precisely the number of $k$-tuples ($k\geq 2$) of distinct vertices inthis sub-tree, in small generations and satisfying an hereditary condition. Aspecial 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 givethe major contribution to this range. As an application, we study the genealogyof $k\geq 2$ distinct vertices of the tree $\mathcal{R}_T$ picked uniformlyfrom those in small generations. It turns out that two or more vertices amongthem share a common ancestor for the last time in the remote past. We alsopoint out an hereditary character in their genealogical tree due to the randomenvironment.

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