Current steady state rolling tire calculations often do not include treads because treads
destroy the rotational symmetry of the tire. We describe two methodologies to compute time
periodic solutions of a two-dimensional viscoelastic tire with treads: solving a minimization
problem and solving a system of equations. We also expand on work by Oden and Lin on
free spinning rolling elastic tires in which they disovered a hierachy of N-peak steady state
standing wave solutions. In addition to discovering a two-dimensional hierarchy of standing
wave solutions that includes their N-peak hiearchy, we consider the effects of viscoelasticity
on the standing wave solutions. Finally, a commonplace model of viscoelasticity used in our
numerical experiments led to non-physical elastic energy growth for large tire speeds. We
show that a viscoelastic model of Govindjee and Reese remedies the problem.