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In this paper we study branched coverings of metrized, simplicial trees $F:T\to T$ which arise from polynomial maps $f:ℂ\to ℂ$ with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space $\mathbb{P}{T}_D$ 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 ${\Delta }^{*}$ can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial...