Combinatorial configurations of fibered polynomials

We study from a combinatorial point of view quadratic fibered polynomials. The main object of this paper is the construction of a combinatorial model of a subset of the parameter space. This set (the configuration space) is proved to be a Hausdorff compact and connected space. Then, we prove that an abstract configuration which is non-recurrent can be uniquely realized by a quadratic fibered polynomial.