In this paper we present a tree structure for storing points from a normed space whose norm is effectively computable. We then give an algorithm for finding the nearest point from the tree to a given query point. Our algorithm searches the tree and uses the triangle inequality to eliminate searching of the entirety of some branches of the tree whenever a certain predicate is satisfied. Our data structure uses 0(n) for storage. Empirical data which we have gathered suggest that the expected complexity for preprocessing and the search time are, respectively, 0(nlogn) and 0(logn).