The k-ball l-path branch weight centroid

If x is a vertex of a tree T of radius r, if k and l are integers, if 0 ⩽ k ⩽ r, 0⩽ l ⩽ r, and if P is an l-path with one end at x, then define β(x; k, P) to be the number of vertices of T that are reachable from x via the l-path P and that are outside of the k-ball about x. That is, β(x;k,P) = ∣{yϵV(T):y is reachable from x via P,d(x,y) > k}∣. Define the k-ball l-path branch weight of x, denoted β(x;k,l), to be max {β(x;k,P):P an l-path with one end at x}, and define the k-balll-path branch weight centroid of T, denoted B(T;k,l), to be the set xϵV(T): β(x;k,l) ⩽ β(y;k,l), yϵV(T). This two-parameter family of central sets in T includes the one-parameter family of central sets called the k-nuclei introduced by Slater (1981) which has been shown to be the one parameter family of central sets called the k-branch weight centroids by Zaw Win (1993). It also includes the one-parameter family of central sets called the k-ball branch weight centroid introduced by Reid (1991). In particular, this new family contains the classical central sets, the center and the median (which Zelinka (1968) showed is the ordinary branch weight centroid). The sets obtained for particular values of k and l are examined, and it is shown that for many values they consist of one vertex or two adjacent vertices.