  The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM)
amplitudes induces a refined graphic expansion, by which any tree-level EYM
amplitude can be expressed as a summation over all possible refined graphs.
Each graph contributes a unique coefficient as well as a proper combination of
color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate
EYM amplitudes through YM amplitudes, the latter have much simpler structures
in four dimensions than the former. In this paper, we classify the refined
graphs for the expansion of EYM amplitudes into $\text{N}^{\,k}$MHV sectors.
Amplitudes in four dimensions, which involve $k+2$ negative-helicity particles,
at most get non-vanishing contribution from graphs in $\text{N}^{\,k'(k'\leq
k)}$MHV sectors. By the help of this classification, we evaluate the
non-vanishing amplitudes with two negative-helicity particles in four
dimensions. We establish a correspondence between the refined graphs for
single-trace amplitudes with $(g^-_i,g^-_j)$ or $(h^-_i,g^-_j)$ configuration
and the spanning forests of the known Hodges determinant form. Inspired by this
correspondence, we further propose a symmetric formula of double-trace
amplitudes with $(g^-_i,g^-_j)$ configuration. By analyzing the cancellation
between refined graphs in four dimensions, we prove that any other tree
amplitude with two negative-helicity particles has to vanish.
