On a conjecture about Wiener index in iterated line graphs of trees

Let G be a graph. Denote by L^i(G) its i-iterated line graph and denote by W(G) its Wiener index. Dobrynin and Melnikov conjectured that there exists no nontrivial tree T and i>=3, such that W(L^i(T))=W(T). We prove this conjecture for trees which are not homeomorphic to the claw K"1","3 and H, where H is a tree consisting of 6 vertices, 2 of which have degree 3.