The computation of the number of limit cycles which appear in an analytic unfolding of planar vector fields is related to the decomposition of the displacement function of this unfolding in an ideal of functions in the parameter space, called the Ideal of Bautin. On the other hand, the asymptotic of the displacement function, for 1-parameter unfoldings of hamiltonian vector fields is given by Melnikov functions which are defined as the coefficients of Taylor expansion in the parameter. It is interesting to compare these two notions and to study if the general estimations of the number of limit cycles in terms of the Bautin ideal could be reduced to the computations of Melnikov functions for some 1-parameter subfamilies.