  While the forward trajectory of a point in a discrete dynamical system is
always unique, in general a point can have infinitely many backward
trajectories. The union of the limit points of all backward trajectories
through $x$ was called by M.~Hero the "special $\alpha$-limit" ($s\alpha$-limit
for short) of $x$. In this article we show that there is a hierarchy of
$s\alpha$-limits of points under iterations of a S-unimodal map: the size of
the $s\alpha$-limit of a point increases monotonically as the point gets closer
and closer to the attractor. The $s\alpha$-limit of any point of the attractor
is the whole non-wandering set. This hierarchy reflects the structure of the
graph of a S-unimodal map recently introduced jointly by Jim Yorke and the
present author.
