  Tree-level scattering amplitudes for gravitons, gluons and Goldstone
particles in any dimensions are strongly constrained by basic principles, and
they are intimately related to each other via various relations. We study two
types of "universal expansions" with respect to gauge bosons and Goldstone
bosons: the former express tree amplitudes in Einstein gravity (Yang-Mills) as
linear combinations of single-trace Einstein-Yang-Mills (Yang-Mills-$\phi^3$)
amplitudes with coefficients given by Lorentz products of polarizations and
momenta; the latter express tree amplitudes in non-linear sigma model,
(Dirac-)Born-Infeld and a special Galileon theory, as linear combinations of
single-trace mixed amplitudes with particles of lower "degree of Adler's zero"
and coefficients given by products of Mandelstam variables. We trace the origin
of gauge-theory expansions to the powerful uniqueness theorem based on gauge
invariance, and expansions in effective field theories can be derived from
gauge-theory ones via a special dimension reduction.
