  In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a
random partially ordered set of points in the unit square,all edges must
respect the ``coordinatewise'' partial order and there must be a directed path
from each vertex to a minimal element. We study the asymptotic behaviour of the
total length of this graph with power weighted edges. The limiting distribution
is given by the sum of a normal component away from the boundary and a
contribution introduced by the boundary effects, which can be characterized by
a fixed point equation, and is reminiscent of limits arising in the
probabilistic analysis of certain algorithms. As the exponent of the power
weighting increases, the distribution undergoes a phase transition from the
normal contribution being dominant to the boundary effects dominating. In the
critical case where the weight is simple Euclidean length, both effects
contribute significantly to the limit law. We also give a law of large numbers
for the total weight of the graph.
