  For a finite group G and a finite-dimensional G-module V, we prove a general
result on the Poincar\'e series for the G-invariants in the tensor algebra
T(V). We apply this result to the finite subgroups G of the 2-by-2 special
unitary matrices and their natural module V of 2-by-1 column vectors. Because
these subgroups are in one-to-one correspondence with the simply laced affine
Dynkin diagrams by the McKay correspondence, the Poincar\'e series obtained are
the generating functions for the number of walks on the simply laced affine
Dynkin diagrams.
