A description of how to define in a mathematically rigorous way the beta function, well known from the nonrigorous perturbative formulation of quantum field theory. The definition is based on a kind of effective potentials, for which a "tree expansion'' is proposed. It is also described how to use such a beta function in order to rigorously construct a model which is renormalizable and asymptotically free (but not super-renormalizable model). The paper also contains a discussion of applicability of the tree expansion in statistical mechanics.