  The abstract tile assembly model (aTam) is a model of DNA self-assembly. Most
of the studies focus on cooperative aTam where a form of synchronization
between the tiles is possible. Simulating Turing machines is achievable in this
context. Few results and constructions are known for the non-cooperative case
(a variant of Wang tilings where assemblies do not need to cover the whole
plane and some mismatches may occur).
  Introduced by P.E. Meunier and D. Regnault, efficient paths are a non-trivial
construction for non-cooperative aTam. These paths of width nlog(n) are
designed with n different tile types. Assembling them relies heavily on a form
of ``non-determinism''. Indeed, the set of tiles may produced different finite
terminal assemblies but they all contain the same efficient path. Directed
non-cooperative aTam does not allow this non-determinism as only one assembly
may be produced by a tile assembly system. This variant of aTam is the only one
who was shown to be decidable.
  In this paper, we show that if the terminal assembly of a directed
non-cooperative tile assembly system is finite then its width and length are of
linear size according to the size of the tile assembly system. This result
implies that the construction of efficient paths cannot be generalized to the
directed case and that some computation must rely on a competition between
different paths. It also implies that the construction of a square of width n
using 2n-1 tiles types is asymptotically optimal. Moreover, we hope that the
techniques introduced here will lead to a better comprehension of the
non-directed case.
