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 σ T ( v ) the sum of distances from a vertex v to all other vertices in T . Barefoot et al. (1997) determined extremal values of σ T ( w ) / σ T ( u ) , σ T ( w ) / σ T ( v ) , σ ( T ) / σ T ( v ) , and σ ( T ) / σ T ( 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 F T ( 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 F T ( w ) / F T ( u ) , F T ( w ) / F T ( v ) . In this paper we characterize the extremal values of F ( T ) / F T ( v ) , and F ( T ) / F T ( 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.