Abstract. Let G be a graph. Denote by Lⁱ(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(Lⁱ(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, two of which have degree 3.