  We consider the Directed Spanning Forest (DSF) constructed as follows: given
a Poisson point process N on the plane, the ancestor of each point is the
nearest vertex of N having a strictly larger abscissa. We prove that the DSF is
actually a tree. Contrary to other directed forests of the literature, no
Markovian process can be introduced to study the paths in our DSF. Our proof is
based on a comparison argument between surface and perimeter from percolation
theory. We then show that this result still holds when the points of N
belonging to an auxiliary Boolean model are removed. Using these results, we
prove that there is no bi-infinite paths in the DSF.
