The basin of infinity of a polynomial map f : {\bf C} \arrow {\bf C}
carries a natural foliation and a flat metric with singularities, making it
into a metrized Riemann surface X(f). As f diverges in the moduli space of
polynomials, the surface X(f) collapses along its foliation to yield a
metrized simplicial tree (T,η), with limiting dynamics F : T \arrow T.
In this paper we characterize the trees that arise as limits, and show they
provide a natural boundary \PT_d compactifying the moduli space of
polynomials of degree d. We show that (T,η,F) records the limiting
behavior of multipliers at periodic points, and that any divergent meromorphic
family of polynomials \{f_t(z) : t \mem \Delta^* \} can be completed by a
unique tree at its central fiber. Finally we show that in the cubic case, the
boundary of moduli space \PT_3 is itself a tree.
The metrized trees (T,η,F) provide a counterpart, in the setting of
iterated rational maps, to the R-trees that arise as limits of
hyperbolic manifolds.Comment: 60 page
