Appears in collection : Combinatorics and Arithmetic for Physics: special days

We here introduce some combinatorial and analytic tools, conceived to make possible to perform new expansions in the context of constructive field theory and multiscale analysis. These formulas generalize the idea of performing cluster expansion using a sum indexed by forest to the case of a Taylor expansion of order more than zero. They are expected to help construct new field theories of the matrices and tensors type.