  We present a supersymmetric generalization of the MHV vertex expansion for
all tree amplitudes in N=4 SYM theory. In addition to the choice of a reference
spinor, this super MHV vertex expansion also depends on four reference
Grassmann parameters. We demonstrate that a significant fraction of diagrams in
the expansion vanishes for a judicious choice of these Grassmann parameters,
which simplifies the computation of amplitudes. Even pure-gluon amplitudes
require fewer diagrams than in the ordinary MHV vertex expansion.
  We show that the super MHV vertex expansion arises from the recursion
relation associated with a holomorphic all-line supershift. This is a
supersymmetric generalization of the holomorphic all-line shift recently
introduced in arXiv:0811.3624. We study the large-z behavior of generating
functions under these all-line supershifts, and find that they generically
provide 1/z^k falloff at (Next-to)^k MHV level. In the case of anti-MHV
generating functions, we find that a careful choice of shift parameters
guarantees a stronger 1/z^(k+4) falloff. These particular all-line supershifts
may therefore play an important role in extending the super MHV vertex
expansion to N=8 supergravity.
