  We consider non cooperative binding in so called `temperature 1', in
deterministic (here called {\it confluent}) tile self-assembly systems (1-TAS)
and prove the standing conjecture that such systems do not have universal
computational power. We call a TAS whose maximal assemblies contain at least
one ultimately periodic assembly path {\it para-periodic}. We observe that a
confluent 1-TAS has at most one maximal producible assembly, $\alpha_{max}$,
that can be considered a union of path assemblies, and we show that such a
system is always para-periodic. This result is obtained through a superposition
and a combination of two paths that produce a new path with desired properties,
a technique that we call \emph{co-grow} of two paths. Moreover we provide a
characterization of an $\alpha_{max}$ of a confluent 1-TAS as one of two
possible cases, so called, a grid or a disjoint union of combs. To a given
$\alpha_{max}$ we can associate a finite labeled graph, called \emph{quipu},
such that the union of all labels of paths in the quipu equals $\alpha_{max}$,
therefore giving a finite description for $\alpha_{max}$. This finite
description implies that $\alpha_{max}$ is a union of semi-affine subsets of
$\mathbb{Z}^2$ and since such a finite description can be algorithmicly
generated from any 1-TAS, 1-TAS cannot have universal computational power.
