  Consider the d-dimensional lattice 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 dth
co-ordinates of v and w satisfy w(d)=v(d)-1. In case of nonuniqueness 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 3 and it is an infinite collection of
distinct trees for d\geq4. In addition, for any dimension, we show that there
is no bi-infinite path in the tree and we also obtain central limit theorems of
(a) the number of vertices of a fixed degree \nu and (b) the number of edges of
a fixed length l.
