  We study a system of coalescing random walks on the integer lattice
$\mathbb{Z}^{d}$ in which the walk is oriented in the $d$-th direction and
follows certain specified rules. We first study the geometry of the paths and
show that, almost surely, the paths from a graph consisting of just one tree
for dimentions $d=2,3$ and infinitely many disjoint trees for dimensions $d\geq
4$. Also, there is no bi-infinite path in the graph almost surely for $d\geq
2$. Subsequently, we prove that for $d=2$ the diffusive scaling of this system
converges in distribution to the Brownian web.
