  Balogh, Bollob\'{a}s and Morris (2006) have described a threshold phenomenon
in the behavior of the profile of hereditary classes of ordered graphs. In this
paper, we give an other look at their result based on the notion of monomorphic
decomposition of a relational structure introduced in \cite{P-T-2013}. We prove
that the class $\mathfrak S$ of ordered binary structures which do not have a
finite monomorphic decomposition has a finite basis (a subset $\mathfrak A$
such that every member of $\mathfrak S$ embeds some member of $\mathfrak A$).
In the case of ordered reflexive directed graphs, the basis has 1242 members
and the profile of their ages grows at least as the Fibonacci function. From
this result, we deduce that the following dichotomy property holds for every
hereditary class $\mathfrak C$ of finite ordered binary structures of a given
finite type. Either there is an integer $\ell$ such that every member of
$\mathfrak C$ has a monomorphic decomposition into at most $\ell$ blocks and in
this case the profile of $\mathfrak C$ is bounded by a polynomial of degree
$\ell-1$ (and in fact is a polynomial), or $\mathfrak C$ contains the age of a
structure which does not have a finite monomorphic decomposition, in which case
the profile of $\mathfrak C$ is bounded below by the Fibonacci function.
