We study the asymptotic properties of a minimal spanning tree formed
by $n$ points uniformly distributed in the unit square, where the
minimality is amongst all rooted spanning trees with a direction of
growth. We show that the number of branches from the root of this
tree, the total length of these branches, and the length of the
longest branch each converge weakly. This model is related to the
study of record values in the theory of extreme value statistics and
this relation is used to obtain our results. The results also hold
when the tree is formed from a Poisson point process of intensity $n$
in the unit square.