Wiener index of binomial trees and Fibonacci trees

Given a simple connected undirected graph G = (V , E), the Wiener index W (G) of G is defined as half the sum of the distances d(u, v) between all pairs of vertices u, v of G, where d(u, v) denotes the distance (the number of edges on a shortest path between u and v) between u, v in G. We obtain an expression for W (G), where G is a binomial tree. For Fibonacci trees and binary Fibonacci trees with Fk (k-th Fibonacci number) vertices, we outline algorithms for computing their Wiener index in time O(logFk). This may be compared with the existing result: for any tree T with n vertices, W (T ) can be computed by an algorithm in time O(n).