  We prove that all open string theory disc amplitudes in a flat background
obey Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion relations, up to a
possible reality condition on a kinematic invariant. Arguments that the same
holds for tree level closed string amplitudes are given as well. Non-adjacent
BCFW-shifts are related to adjacent shifts through monodromy relations for
which we provide a novel CFT based derivation. All possible recursion relations
are related by old-fashioned string duality. The field theory limit of the
analysis for amplitudes involving gluons is explicitly shown to be smooth for
both the bosonic string as well as the superstring. In addition to a proof a
less rigorous but more powerful argument based on the underlying CFT is
presented which suggests that the technique may extend to a much more general
setting in string theory. This is illustrated by a discussion of the open
string in a constant B-field background and the closed string on the level of
the sphere.
