  The aim of this work is to outline in some detail the use of combinatorial
algebra in planar quantum field theory. Particular emphasis is given to the
relations between the different types of planar Green's functions. The key
object is a Hopf algebra which is naturally defined on non-commuting sources,
and the fact that its genuine unshuffle coproduct splits into left- and right
unshuffle half-coproduts. The latter give rise to the notion of unshuffle
bialgebra. This setting allows to describe the relation between planar full and
connected Green's functions by solving a simple linear fixed point equation. A
modification of this linear fixed point equation gives rise to the relation
between planar connected and one-particle irreducible Green's functions. The
graphical calculus that arises from this approach also leads to a new
understanding of functional calculus in planar QFT, whose rules for
differentiation with respect to non-commuting sources can be translated into
the language of growth operations on planar rooted trees. We also include a
brief outline of our approach in the framework of non-planar theories.
