On the number of leaves of a Euclidean minimal spanning tree.

Let V sub (k,n) be the number of vertices of degree k in the Euclidean minimal spanning tree of X sub i, 1 = i = n, where the X sub i are independent, absolutely continuous random variables with values in R sup d. It is proved that n sup (-1) V sub (k,n) converges with probability one to a constant alpha sub (k,d). Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of V sub (k,n).