Enumeration of subtrees of trees

Let T be a weighted tree. The weight of a subtree T1 of T is defined as the product of weights of vertices and edges of T1. We obtain a linear-time algorithm to count the sum of weights of subtrees of T. As applications, we characterize the tree with the diameter at least d, which has the maximum number of subtrees, and we characterize the tree with the maximum degree at least Δ, which has the minimum number of subtrees.