  We give the path model solution for the cluster algebra variables of the
$A_r$ $T$-system with generic boundary conditions. The solutions are partition
functions of (strongly) non-intersecting paths on weighted graphs. The graphs
are the same as those constructed for the $Q$-system in our earlier work, and
depend on the seed or initial data in terms of which the solutions are given.
The weights are "time-dependent" where "time" is the extra parameter which
distinguishes the $T$-system from the $Q$-system, usually identified as the
spectral parameter in the context of representation theory. The path model is
alternatively described on a graph with non-commutative weights, and cluster
mutations are interpreted as non-commutative continued fraction rearrangements.
As a consequence, the solution is a positive Laurent polynomial of the seed
data.
