  The task of Monte Carlo simulation of the evolution of the parton
distributions in QCD and of constructing new parton shower Monte Carlo
algorithms requires new way of organizing solutions of the QCD evolution
equations, in which quark-gluon transitions on one hand and quark-quark or
gluon-gluon transitions (pure gluonstrahlung) on the other hand, are treated
separately and differently. This requires certain reorganization of the
iterative solutions of the QCD evolution equations and leads to what we refer
to as a hierarchic iterative solutions of the evolution equations. We present
three formal derivations of such a solution. Results presented here are already
used in the other recent works to formulate new MC algorithms for the
parton-shower-like implementations of the QCD evolution equations. They are
primarily of the non-Markovian type. However, such a solution can be used for
the Markovian-type MCs as well. We also comment briefly on the relation of the
presented formalism to similar methods used in other branches of physics.
