  Since the Connes--Kreimer Hopf algebra was proposed, revisiting present
quantum field theory has become meaningful and important from algebraic points.
In this paper, the Hopf algebra in the cutting rules is constructed. Its
coproduct contains all necessary ingredients for the cutting equation crucial
to proving perturbative unitarity of the S-matrix. Its antipode is compatible
with the causality principle. It is obtained by reducing the Hopf algebra in
the largest time equation which reflects partitions of the vertex set of a
given Feynman diagram. First of all, the Connes--Kreimer Hopf algebra in the
BPHZ renormalization instead of the dimensional regularization and the minimal
subtraction is described so that the strategy of setting up Hopf algebraic
structures of Feynman diagrams becomes clear.
