  The two-dimensional directed spanning forest (DSF) introduced by Baccelli and
Bordenave is a planar directed forest whose vertex set is given by a
homogeneous Poisson point process $\mathcal{N}$ on $\mathbb{R}^2$. If the DSF
has direction $-e_y$, the ancestor $h(u)$ of a vertex $u \in \mathcal{N}$ is
the nearest Poisson point (in the $L_2$ distance) having strictly larger
$y$-coordinate. This construction induces complex geometrical dependencies. In
this paper we show that the collection of DSF paths, properly scaled, converges
in distribution to the Brownian web (BW). This verifies a conjecture made by
Baccelli and Bordenave in 2007.
