Minimal spanning tree: a surprising property of the center.
Preprint
It is known since the work of C. Jordan in 1869 that the center of any tree, the set of middle vertices of any longest path of the tree consists of one vertex or two adjacent vertices in the tree. This paper is concerned with the stability of the center in space: If Xn = {X1, X2,..., Xn} denotes n independent random points drawn uniformly from the unit d-cube Bd= [0,1]d and cent (Xn) denotes the center of the minimal spanning tree on Xn, is cent(Xn) a consistent estimator for the center of Bd ? While the answer is ``Yes" for d=1, perhaps surprisingly, this question is answered negatively here for the uniform and related Poisson point process models for d \geq 2. It is shown that the non-intersection probability of cent(Xn) with the midcube of Bd centered at (1/2,..., 1/2) of side a for 0 < a \leq 1/3 is at least 1/8 for every positive integer n. Our approach engages the longest edge length in a ``circuit" and the numbers of edges of two paths of a ``semi-circuit" that form when two minimal spanning trees on point sets on two disjoint regions in Rd are merged.