Extending some properties from the Euclidean plane to any normed plane, we show the validity of the Monma-Paterson-Suri-Yao algorithm for finding the maximum-weighted spanning tree of a set of $n$ points, where the weight of an edge is the distance between the end points measured by the norm and there are not repeated distances. For strictly convex normed planes, we expose an strategy for moving slightly the points of the set in order to obtain distinct distances.