This paper considers the asymptotic distribution of the longest edge of the minimal spanning tree and nearest neighbor graph on X1,…,XNn where X1,X2,… are i.i.d. in ℜ2 with distribution F and Nn is independent of the Xi and satisfies Nn/n→p1. A new approach based on spatial blocking and a locally orthogonal coordinate system is developed to treat cases for which F has unbounded support. The general results are applied to a number of special cases, including elliptically contoured distributions, distributions with independent Weibull-like margins and distributions with parallel level curves.