  Tree level multi-trace Yang-Mills-scalar (YMS) amplitudes have been shown to
satisfy a recursive expansion formula, which expresses any YMS amplitude by
those with fewer gluons and/or scalar traces. In an earlier work, the
single-trace expansion formula has been shown to be determined by the
universality of soft behavior. This approach is nevertheless not extended to
multi-trace case in a straightforward way. In this paper, we derive the
expansion formula of tree-level multi-trace YMS amplitudes in a bottom-up way:
we first determine the simplest amplitude, the double-trace pure scalar
amplitude which involves two scalars in each trace. Then insert more scalars to
one of the traces. Based on this amplitude, we further obtain the double-soft
behavior when the trace containing only two scalars is soft. The multi-trace
amplitudes with more scalars and more gluons finally follow from the
double-soft behavior as well as the single-soft behaviors which has been
derived before.
