  In this paper, we have introduced a fundamentally different approach, based
on a bottom-up methodology, to expand tree-level Yang-Mills (YM) amplitudes
into Yang-Mills-scalar (YMS) amplitudes and Bi-adjoint-scalar (BAS) amplitudes.
Our method relies solely on the intrinsic soft behavior of external gluons,
eliminating the need for external aids such as Feynman rules or CHY rules. The
recursive procedure consistently preserves explicit gauge invariance at every
step, ultimately resulting in a manifest gauge-invariant outcome when the
initial expression is already framed in a gauge-invariant manner. The resulting
expansion can be directly analogized to the expansions of gravitational (GR)
amplitudes using the double copy structure. When combined with the expansions
of Einstein-Yang-Mills amplitudes obtained using the covariant double copy
method from existing literature, the expansions presented in this note yield
gauge-invariant BCJ numerators.
