Centroids to centers in trees

If k is a nonnegative integer and x is a vertex of a tree T , the k ‐ball branch weight of x , denoted b(x;k) , is the number of vertices in a largest subtree of T , all of whose vertices are a distance at least k + 1 from x . The k ‐ball branch weight centroid of T , denoted W(T;k) , consists of all vertices x of T for which b(x;k) is a minimum. The usual branch weight centroid of T (which is also the median) is W ( T ;0), and the center of T is W(T;r) , where r is the radius of T . In this paper, the structure of the subgraph spanned by W(T;k) is examined. Similarities and differences with Slater's k ‐centrum [6] and k ‐nucleus [7] are discussed.