  We develop a graphical calculus for the microformal or thick morphisms
introduced by Ted Voronov. This allows us to write the infinite series arising
from pullbacks, compositions, and coordinate transformations of thick morphisms
as sums over bipartite trees. The methods are inspired by those employed by
Cattaneo-Dherin-Felder in their work on formal symplectic groupoids. We also
extend this calculus to quantum thick morphisms, which are special types of
Fourier integral operators quantizing classical thick morphisms. The
relationship between the calculi for classical and quantum thick morphisms
resembles the relationship between the semi-classical and full perturbative
expansions over Feynman diagrams in quantum field theory.
