A method of deriving power series expansions for the correlation functions of lattice systems is described. The concept of lattice constants for rooted graphs is introduced and it is shown how the correlation functions can be expanded in terms of the lattice constants for connected graphs only. The weight functions in the expansions depend on correlation functions for finite systems and may, therefore, be determined by computer methods. This work is an extension of a method previously developed for the free energy. Applications to spin systems, percolation problems and the lattice gas are considered and sum rules for the weight functions are given.