The longest edge of the random minimal spanning tree

For n points placed uniformly at random on the unit square, suppose Mn (respectively, M.n) denotes the longest edge-length of the nearest neighbor graph (respectively, the minimal spanning tree) on these points. It is known that the distribution of n.M2n.logn converges weakly to the double exponential; we give a new proof of this. We show that P[M.n=Mn].1, so that the same weak convergence holds for M.n .