  We show that at tree level, on-shell gauge invariance can be used to fully
determine the first subleading soft-gluon behavior and the first two subleading
soft-graviton behaviors. Our proofs of the behaviors for n-gluon and n-graviton
tree amplitudes are valid in D dimensions and are similar to Low's proof of
universality of the first subleading behavior of photons. In contrast to
photons coupling to massive particles, in four dimensions the soft behaviors of
gluons and gravitons are corrected by loop effects. We comment on how such
corrections arise from this perspective. We also show that loop corrections in
graviton amplitudes arising from scalar loops appear only at the second soft
subleading order. This case is particularly transparent because it is not
entangled with graviton infrared singularities. Our result suggests that if we
set aside the issue of infrared singularities, soft-graviton Ward identities of
extended BMS symmetry are not anomalous through the first subleading order.
