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Using the Cachazo-He-Yuan (CHY) formalism, we prove a recursive expansion oftree level single trace Einstein-Yang-Mills (EYM) amplitudes with arbitrarynumber of gluons and gravitons, which is valid for general spacetime dimensionsand any helicity configurations. The recursion is written in terms offewer-graviton EYM amplitudes and pure Yang-Mills (YM) amplitudes, which can befurther carried out until we reach an expansion in terms of pure YM amplitudesin Kleiss-Kuijf (KK) basis. Our expansion then generates naturally a spanningtree structure rooted on gluons whose vertices are gravitons. We furtherpropose a set of graph theoretical rules based on spanning trees that evaluatedirectly the pure YM expansion coefficients.

Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame