Altered Wiener indices of thorn trees

Three modifications of the Wiener index W(G) of a structure G have been recently proposed: the lambda-modified Wiener index W-lambda(G) = Sigma(uv epsilon E(G)) n(G)(u,v)(lambda)n(G)(v,u)(lambda), the lambda-variable Wiener index lambda W(G)=1/2 Sigma(uv epsilon E(G)) (n(G)(lambda) - n(G)(u,v)(lambda) - n(G)(v,u)(lambda)) and the lambda-altered Wiener index W-min,W-lambda(G) = 1/2 Sigma(uv epsilon E(G)) (n(G)(lambda) m(G)(u,v)(lambda) - m(G)(u,v)(2 lambda)) where n(G) is the number of vertices of G, and m(G) = min {n(G)(u,v), n(G)(v,u)}. For a given positive integer k, explicit formulae are available for calculating the k-modified Wiener index and the k-variable Wiener index of a thorn tree by means of the i-modified Wiener indices and the i-variable Wiener indices, respectively, of the parent tree for integers i and k with 0 <= i <= k. It is pointed out in the present report that this is not the case of the k-altered Wiener index.