  In this paper, we investigate the expansion of tree level multitrace
Einstein-Yang-Mills (EYM) amplitudes. First, we propose two types of recursive
expansions of tree level EYM amplitudes with an arbitrary number of gluons,
gravitons and traces by those amplitudes with fewer traces or/and gravitons.
Then we give many support evidence, including proofs using the Cachazo-He-Yuan
(CHY) formula and Britto-Cachazo-Feng-Witten (BCFW) recursive relation. As a
byproduct, two types of generalized BCJ relations for multitrace EYM are
further proposed, which will be useful in the BCFW proof. After one applies the
recursive expansions repeatedly, any multitrace EYM amplitudes can be given in
the Kleiss-Kuijf (KK) basis of tree level color ordered Yang-Mills (YM)
amplitudes. Thus the Bern-Carrasco-Johansson (BCJ) numerators, as the expansion
coefficients, for all multitrace EYM amplitudes are naturally constructed.
