  We consider idealized parton shower event generators that treat parton spin
and color exactly, leaving aside the choice of practical approximations for
spin and color. We investigate how the structure of such a parton shower
generator is related to the structure of QCD. We argue that a parton shower
with splitting functions proportional to $\alpha_s$ can be viewed not just as a
model, but as the lowest order approximation to a shower that is defined at any
perturbative order. To support this argument, we present a formulation for a
parton shower at order $\alpha_s ^k$ for any $k$. Since some of the input
functions needed are specified by their properties but not calculated, this
formulation does not provide a useful recipe for an order $\alpha_s ^k$ parton
shower algorithm. However, in this formulation we see how the operators that
generate the shower are related to operators that specify the infrared
singularities of QCD.
