On The Wiener Index And The Subtree Counting Of Trees

Wiener index (the subtree counting) attracts the researchers as follows: how the Wiener index (the subtree counting) depends on the structure of the tree, how to calculate the Wiener index (the subtree counting) effectively, the tree with the largest and the smallest Wiener index (the subtree counting), and the order of the Wiener index (the subtree counting) under certain conditions.We study the relationship of the subtree and the generating function of the tree. We give the trees with the second smallest and the second largest subtrees by the study of transforing a pendant edge.The star tree reduces the Wiener index while the path tree increases the Wiener index. We study the order of some tree by the Wiener index, We give the structure of the trees with the smallest and the largest Wiener index under certain conditions.We study the double stars tree and the extended double stars tree the properties of the Wiener index and the subtrees, and we give the trees with the maximum Wiener index and the minimum subtrees. And we give out the relationship of the Wiener index and the subtrees under the transformation of the pendant edge.