The degree d \ge 2 shift locus S_d consists of affine conjugacy classes of complex polynomials whose critical points all escape to infinity under iterations of the polynomial. We relate generic points in the shift locus S_d to metric graphs. Using thermodynamic metrics on the space of metric graphs/Culler-Vogtmann outer space, we obtain a distance function on S_d. We show that S_d is incomplete if and only if d \ge 3 and the metric completion contains a subset homeomorphic to the space of trees PST_d^* introduced by DeMarco-Pilgrim. This is joint work with Hongming Nie.

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