We obtain some formulae for calculation of the coefficients of four special types of terms in τ 2 k , k = 1, 2, ... (1-1 corresponding to four type of (2k + 1)-vertex free unlabeled trees, k = 1, 2, ..., respectively), for a fixed step size τ, in the tree-expansion of the formal energy of the mid-point rule. And, we give an estimate of the difference between the formal energy H and the standard Hamiltonian H in some domain Ω under the assumptions (i)|H is smooth and bounded in Ω, and(ii)|the absolute values of the coefficients of the terms in τ 2 k are uniformly bounded by ησ 2 k for some constants η =< 1, σ > 0 and for any k =< 1.