  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 $(T,\eta)$, 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 $(T,\eta,F)$ 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 $(T,\eta,F)$ provide a counterpart, in the setting of
iterated rational maps, to the ${\bf R}$-trees that arise as limits of
hyperbolic manifolds.
