  Consider the $d$ dimensional lattice $\mathbb{Z}^d$ where each vertex is open
or closed with probability $p$ or $1-p$ respectively. An open vertex
$\mathbb{u} := (\mathbb{u}(1), \mathbb{u}(2),...,\mathbb{u}(d))$ is connected
by an edge to another open vertex which has the minimum $L_1$ distance among
all the open vertices with $\mathbb{x}(d)>\mathbb{u}(d)$. It is shown that this
random graph is a tree almost surely for $d=2$ and 3 and it is an infinite
collection of disjoint trees for $d\geq 4$. In addition for $d=2$, we show that
when properly scaled, family of its paths converges in distribution to the
Brownian web.
