  We make a full classification of scalar monomials built of the Riemann
curvature tensor up to the quadratic order and of the covariant derivatives of
the scalar field up to the third order. From the point of view of the effective
field theory, the third or even higher order covariant derivatives of the
scalar field are of the same order as the higher curvature terms, and thus
should be taken into account. Moreover, higher curvature terms and higher order
derivatives of the scalar field are complementary to each other, of which novel
ghost-free combinations may exist. We make a systematic classification of all
the possible monomials, according to the numbers of Riemann tensor and higher
derivatives of the scalar field in each monomial. Complete basis of monomials
at each order are derived, of which linear combinations may yield novel
ghost-free Lagrangians. We also develop a diagrammatic representation of the
monomials, which may help to simplify the analysis.
