  We compute the tree-level $H^2R^3$ couplings of type II strings and provide
some basic tests of the couplings by considering both K3 and Calabi-Yau
threefold compactifications. Curiously, additional kinematical structures show
up at tree level that are not present in the one-loop couplings. This has
interesting implications for type II supersymmetry as well as $SL(2, \mathbb
Z)$ duality in type IIB strings.
