  We present an analogue of the differential calculus in which the role of
polynomials is played by certain ordered sets and trees. Our combinatorial
calculus has all nice features of the usual calculus and has an advantage that
the elements of the considered ordered sets might carry some additional
information. In this way an analytic proof of generalized Cauchy identities
from the previous work of the second author can be directly reformulated in our
new language of the combinatorial calculus; furthermore the additional
information carried by vertices determines uniquely the bijections presented in
Part I of this series.
