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 $ v$ is connected by an edge to the closest open vertex $
w$ such that the $d$th co-ordinates of $ v$ and $ w$ satisfy ${w}(d) =
v(d) -1$. In case of non-uniqueness of such a vertex $w$, we choose
any one of the closest vertices with equal probability and
independently of the other random mechanisms. It is shown that this
random graph is a tree almost surely for $d=2$ and it is a infinite
collection of distinct trees for $d \geq 4$. In addition, for any
dimension, we obtain central limit theorems of (a) the number of
vertices of a fixed degree $\nu$ and (b) of the number of edges of a
fixed length $l$. These results are obtained by using the martingale
convergence theorem and a coupling of the process with independent
random walks.