Trees and the dynamics of polynomials

In this paper we study branched coverings of metrized, simplicial trees F : T → T which arise from polynomial maps f : ℂ → ℂ with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space PT_D compactifying the moduli space of polynomials of degree D; that F records the asymptotic behavior of the multipliers of f; and that any meromorphic family of polynomials over Δ* can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial enumeration of the trees that arise, and show that ℙT₃ is itself a tree.