  We extend on-shell bootstrap methods from spacetime amplitudes to the
worldsheet objects of the CHY formalism. We find that the integrands
corresponding to tree-level non-linear sigma model, Yang-Mills and $(DF)^2$
theory are determined by demanding enhanced UV scaling under BCFW shifts. Back
in spacetime, we also find that $(DF)^2$ theory is fixed by gauge invariance/UV
scaling and simple locality assumptions.
