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We present an analogue of the differential calculus in which the role ofpolynomials is played by certain ordered sets and trees. Our combinatorialcalculus has all nice features of the usual calculus and has an advantage thatthe elements of the considered ordered sets might carry some additionalinformation. In this way an analytic proof of generalized Cauchy identitiesfrom the previous work of the second author can be directly reformulated in ournew language of the combinatorial calculus; furthermore the additionalinformation carried by vertices determines uniquely the bijections presented inPart I of this series.

Generalized Cauchy identities, trees and multidimensional Brownian motions. Part II: Combinatorial differential calculus