  The Brownian web can be roughly described as a family of coalescing
one-dimensional Brownian motions starting at all times in $\R$ and at all
points of $\R$. It was introduced by Arratia; a variant was then studied by
Toth and Werner; another variant was analyzed recently by Fontes, Isopi, Newman
and Ravishankar. The two-dimensional \emph{Poisson tree} is a family of
continuous time one-dimensional random walks with uniform jumps in a bounded
interval. The walks start at the space-time points of a homogeneous Poisson
process in $\R^2$ and are in fact constructed as a function of the point
process. This tree was introduced by Ferrari, Landim and Thorisson. By
verifying criteria derived by Fontes, Isopi, Newman and Ravishankar, we show
that, when properly rescaled, and under the topology introduced by those
authors, Poisson trees converge weakly to the Brownian web.
