Proof of the MHV vertex expansion for all tree amplitudes in N=4 SYM theory

We prove the MHV vertex expansion for all tree amplitudes of N=4 SYM theory. The proof uses a shift acting on all external momenta, and we show that every N^kMHV tree amplitude falls off as 1/z^k, or faster, for large z under this shift. The MHV vertex expansion allows us to derive compact and efficient generating functions for all N^kMHV tree amplitudes of the theory. We also derive an improved form of the anti-NMHV generating function. The proof leads to a curious set of sum rules for the diagrams of the MHV vertex expansion.