Several Problems About Subtree Number

For a connected tree H, if V(H)(?) V(G) and E(H)(?) E(G) then H is a subtree of G. The total number of subtrees of a graph G is denoted by F(G). The number of subtrees associated with the network reliability, recently, some results were published in JCTB, JTS, AAM, SIAM. These results relate to the enumeration of the subtrees, characterize the extremum graphs and the inverse problems about the number of sub-trees.This paper mainly studies the number of subtrees of unicyclic graph, bicyclic graph and the trees with a given diameter.In the first chapter we define some terminologies that we will require later on and the advances about the number of subtrees.The second chapter deals with a variety of formulas for computing the number of subtrees of trees.The third chapter give the transformations which increase or decrease the number of subtrees, By the transformations, we characterize the first largest, the smallest num-ber of subtrees of unicyclic graph, bicyclic graph, and characterize some trees with a given diameter ordered by their number of subtrees.In the last chapter, we propose some problems for further research on the number of subtrees.