  By the use of cyclic symmetry, KK relations and BCJ relations, one can reduce
the number of independent $N$-point color-ordered tree amplitudes in gauge
theory and string theory from $N!$ to $(N-3)!$. In this paper, we investigate
these relations at tree-level in both string theory and field theory. We will
show that there are two primary relations. All other relations can be generated
by the primary relations. In string theory, the primary relations can be chosen
as cyclic symmetry as well as either the fundamental KK relation or the
fundamental BCJ relation. In field theory, the primary relations can only be
chosen as cyclic symmetry and the fundamental BCJ relation. We will further
show a kind of more general relation which can also be generated by the primary
relations. The general formula of the explicit minimal-basis expansions for
color-ordered open string tree amplitudes will be given and proven in this
paper.
