Motivated by practical VLSI applications, we study the maximum vertex degree in a minimum spanning tree (MST) under arbitrary L p metrics. We show that the maximum vertex degree in a maximum-degree L p MST equals the Hadwiger number of the corresponding unit ball. We then determine the maximum vertex degree in a minimum-degree L p MST; towards this end, we define the MST number, which is closely related to the Hadwiger number. We bound Hadwiger and MST numbers for arbitrary L p metrics, and focus on the L 1 metric, where little was known. We show that the MST number of a diamond is 4, and that for the octahedron the Hadwiger number is 18 and the MST number is either 13 or 14. We also give an exponential lower bound on the MST number for an L p unit ball. Implications to L p minimum spanning trees and related problems are explored.