AbstractIfkis a nonnegative integer andxis a vertex of a treeT, thek‐ball branch weight ofx, denotedb(x;k), is the number of vertices in a largest subtree ofT, all of whose vertices are a distance at leastk+ 1 fromx. Thek‐ball branch weight centroid ofT, denotedW(T;k), consists of all verticesxofTfor whichb(x;k)is a minimum. The usual branch weight centroid ofT(which is also the median) isW(T;0), and the center ofTisW(T;r), whereris the radius ofT. In this paper, the structure of the subgraph spanned byW(T;k)is examined. Similarities and differences with Slater'sk‐centrum [6] andk‐nucleus [7] are di