If you're interested in Delaunay triangulations and Euclidean minimum spanning trees you should check out this MathOverflow question and Bill Thurston's answer to it. The question is: what is the asymptotic behavior of the diameter of an EMST of random points in the unit square? Thurston provides evidence that (unlike a lot of other geometric functionals) the answer isn't proportional or within log factors of proportional to $$\sqrt{n}$$. He also has a fascinating physical analogy involving anti-aircraft radar stations to explain the behavior of distances in random Delaunay triangulations.