  In this note we revisit the problem of explicitly computing tree-level
scattering amplitudes in various theories in any dimension from worldsheet
formulas. The latter are known to produce cubic-tree expansion of tree
amplitudes with kinematic numerators automatically satisfying
Jacobi-identities, once any half-integrand on the worldsheet is reduced to
logarithmic functions. We review a natural class of worldsheet functions called
"Cayley functions", which are in one-to-one correspondence with labelled trees,
and natural expansions of known half-integrands onto them with coefficients
that are particularly compact building blocks of kinematic numerators. We
present a general formula expressing kinematic numerators of all cubic trees as
linear combinations of coefficients of labelled trees, which satisfy Jacobi
identities by construction and include the usual combinations in terms of
master numerators as a special case. Our results provide an efficient
algorithm, which is implemented in a Mathematica package, for computing all
tree amplitudes in theories including non-linear sigma model, special Galileon,
Yang-Mills-scalar, Einstein-Yang-Mills and Dirac-Born-Infeld.
