Extremal values of ratios: distance problems vs. subtree problems in trees

The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374–385; Adv. Appl. Math. 34 (2005), 138–155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80(1997), 37–56] 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 . In this paper we test how far the negative correlation between distances and subtrees go if we look for the extremal values of FT (w)/FT (u), FT (w)/FT (v), 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 u,w are leaves in T—the complete analogue, changing distances to the number of subtrees. We include a number of open problems, shifting the interest towards the number of subtrees in graphs.