## Abstract

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.

Original language | American English |
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Journal | Discrete Mathematics |

Volume | 322 |

DOIs | |

State | Published - May 6 2014 |

## Keywords

- Center
- Centroid
- Distances in trees
- Extremal problems
- Star tree
- Subtree core
- Subtrees of trees
- Tree
- Wiener index

## DC Disciplines

- Education
- Mathematics