Abstract: Let n and d be two positive integers. By B_n,d we denote the graph obtained by identifying an endvertex of path P_d with the center of star S_n-d+1, where n ≥ d + 1. By C_n,d we denote the graph obtained by identifying an endvertex of P_d-1 with the center of Stare S_n-d, and the other endvertex of P_d-1 with the center of S₃ where n ≥ d + 3. By E_n,d,k we denote the graph obtained by identifying the vertex v_k of P(v₁ - v₂ - ··· - v_d+1) with the center of Sn-d. In this paper, we completely characterize all trees T which have diameter at least d(d ≥ 3) and satisfy the following conditions: (i) Z(B_n,d) ≤ Z(T) ≤ Z(E_n,d,3) for n = d + 3; (ii) Z(B_n,d) ≤ Z(T) ≤ Z(C_n,d) for n ≥ d + 4. 

Key words: Hosoya index, tree, diameter