  We prove that the MHV vertex expansion is valid for any NMHV tree amplitude
of N=4 SYM. The proof uses induction to show that there always exists a complex
deformation of three external momenta such that the amplitude falls off at
least as fast as 1/z for large z. This validates the generating function for
n-point NMHV tree amplitudes. We also develop generating functions for anti-MHV
and anti-NMHV amplitudes. As an application, we use these generating functions
to evaluate several examples of intermediate state sums on unitarity cuts of
1-, 2-, 3- and 4-loop amplitudes. In a separate analysis, we extend the recent
results of arXiv:0808.0504 to prove that there exists a valid 2-line shift for
any n-point tree amplitude of N=4 SYM. This implies that there is a BCFW
recursion relation for any tree amplitude of the theory.
