in string theory. The left- and right-moving sectors are connected into a single one. We give particular forms of the relations for the amplitude A (1 − g , 2 + g , ..., i − g , ..., N + g , ( N + 1) + $_{h}$), and A (1 − g , 2 + g , ..., i − g , ..., N + g , ( N + 1) + $_{h}$) , ( N + 2) + $_{h}$. We extend the relation to arbitrary helicity configurations for N + 1 case. The discussions can be extended to arbitrary legs with arbitrary helicity configurations. The tree amplitude with N gluons and M gravitons can be expressed by sum of amplitudes for N + 2 M gluons with appropriate factors. The relation for a given amplitude is not unique, because there are relations among pure-gluon partial amplitudes. Though the D$_{2}$ relations and KLT factorization relations only hold on D$_{2}$ and S$_{2}$ re- spectively in string theory, the field theory limits of both two relations hold in in minimal coupling theory of gauge and gravity. This is because we have two different methods to incorporate gauge degree of freedom in string theory. We would like to thank C. Cao, Y. Q. Wang and Y. Xiao for useful discussions. The work is supported in part by the NNSF of China Grant No. 90503009, No. 10775116, and 973 Program Grant No. 2005CB724508. Here we given the useful properties of spinor helicity formalism[24–26] Positive and neg- ative helicity spinor Acknowledgement Appendix A: Spinor helicity formalism 24 | i $^{±}$〉 ≡  | k ± i 〉 ≡ u$_{±}$ ( k$_{i}$ ) = v$_{∓}$ ( k$_{i}$ ) , 〈 i $^{±}$|  ≡ 〈 k ± i | ≡ ¯ u$_{±}$ ( k$_{i}$ ) = ¯ v$_{±}$ ( k$_{i}$ ) , (A1)