only one leg take negative helicity. In these cases, the D$_{2}$ relations hold trivially. Then we study the tree amplitudes with one and two gravitons in addition to N gluons where N − 2 gluons as well as all the gravitons take positive helicity and two gluons take negative helicity. We will show the D$_{2}$ relations also hold in this case. The discussions can be extended to tree amplitudes where N gluons minimally coupled to M gravitons with arbitrary helicity configurations. In general, any tree amplitudes for gauge-gravity minimal coupling can be expressed by partial tree amplitudes with N + 2 M gluons via D$_{2}$ relations. Our conclusions are given in Section V. Some useful properties of spinor helicity formalism are given in Appendix A. KLT relations[4] in string theory are the relations between amplitudes for closed strings on S$_{2}$ and open string tree amplitudes. KLT relations factorize amplitudes for closed strings on S$_{2}$ into products of two open string tree amplitudes corresponding to the left- and right- moving sectors except for a phase factor 1 4 Where M ( N ) S$_{2}$ is N -point amplitude on S$_{2}$ while A ( N ) and ¯ A ( N ) are the partial tree amplitudes for open strings corresponding to the left- and right-moving sectors. The phase factor only depends on the permutations P and P ′ of the legs in left- and right-moving sectors. The terms in KLT relations can be reduced by contour deformations. In the reduced form, the phase factors become sine functions. After taking the field theory limit α ′ → 0, the KLT 1 In this paper, we use ∼ to omit a proportional factor which does not affect our discussion. II. KLT RELATIONS VERSUS D$_{2}$ RELATIONS M ( N ) S$_{2}$ ∼ κ N − 2 ∑ P,P $_{′}$A ( N $^{)}$( P ) ¯ A ( N $^{)}$( P $^{′}$) e iπF ( P,P $^{$^{′}$)}$, (1)