Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees II

We discovered a dual behavior of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems (Székely and Wang, 2006, 2005). We introduced the concept of subtree core: the subtree core of a tree consists of one or two adjacent vertices of a tree that are contained in the largest number of subtrees. Let σ(T) denote the sum of distances between unordered pairs of vertices in a tree T and ^σT(v) the sum of distances from a vertex v to all other vertices in T. Barefoot et al. (1997) determined extremal values of ^σT(w)/ ^σT(u), ^σT(w)/^σT(v), σ(T)/^σT(v), and σ(T)/^σT(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,w are leaves in T. Let F(T) denote the number of subtrees of T and ^FT(v) the number of subtrees containing v in T. In Part I of this paper we tested how far the negative correlation between distances and subtrees go if we look for (and characterize) the extremal values of ^FT(w)/^FT(u), ^FT(w)/^FT(v). In this paper we characterize the extremal values of F(T)/^FT(v), and F(T)/^FT(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and w is a leaf in T-completing the analogy, changing distances to the number of subtrees.