Dynamics of polynomials with disconnected Julia sets

We study the structure of disconnected polynomial Julia sets. We consider polynomials with an arbitrary number of non-escaping critical points, of arbitrary multiplicity, which interact non-trivially. We use a combinatorial system of a tree with dynamics to give a sufficient condition for the Julia set a polynomial to be an area zero Cantor set. We show that there exist uncountably many combinatorially inequivalent polynomials, which satisfy this condition and have multiple non-escaping critical points, each of which accumulates at all the non-escaping critical points.