  We interpret the degrees which arise in Tevelev's study of scattering
amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection
theory, the boundary geometry of the Hurwitz moduli space yields a simple
recursion for the Tevelev degrees (together with their natural two parameter
generalization). We find exact solutions which specialize to Tevelev's formula
in his cases and connect to the projective geometry of lines and Castelnuovo's
classical count of linear series in other cases. For almost all values, the
calculation of the two parameter generalization of the Tevelev degree is new. A
related count of refined Dyck paths is solved along the way.
