In Bhatt and Roy's minimal directed spanning tree 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 plus 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 being dominant. In the critical case in which the weight is simple Euclidean length, both effects contribute significantly to the limit law.

An algorithm is developed for exact simulation from distributions that are defined as fixed points of maps between spaces of probability measures. The fixed points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixed points with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method.