Corollary 2

If the graph associated with the quiver Γ is a forest, the homomorphism $p_a$ of $G_a$ into $(G/F)_{p(a)}$ is bijective, for every vertex $a$ of G.

In fact, under this assumption, one has Grp(Γ)$_b=\{e_b\}$ for every vertex $b$ of G (Corollary 1 of II, p. 173). It then follows from Proposition 8 that, for every vertex $a$ of G, the group $H_a$ reduces to the identity element.