  In four dimensions, a tree-level double-trace Einstein-Yang-Mills (EYM)
amplitude with two negative-helicity gluons (the $(g^-,g^-)$-configuration)
satisfies a symmetric spanning forest formula, which was derived from the
graphic expansion rule. On another hand, in the framework of Cachazo-He-Yuan
(CHY) formula, the maximally-helicity-violating (MHV) amplitudes are supported
by the MHV solution of scattering equations. The relationship between the
symmetric formula for double-trace amplitudes, and the MHV sector of
Cachazo-He-Yuan (CHY) formula in four dimensions is still not clear. In this
note, we promote a series of transformations of the spanning forests in four
dimensions and then show a systematic way for decomposing the MHV sector of the
CHY formula of double-trace EYM amplitudes. Along this line, the symmetric
formula of double-trace MHV amplitudes is directly obtained by the MHV sector
of CHY formula. We then prove that EYM amplitude with an arbitrary total number
of negative-helicity particles (gravitons and gluons) has to vanish when the
number of negative- (or positive-) helicity gluons is less than the number of
traces.
