  The relation between the spin and the mass of an infinite number of particles
in a $q$-deformed dual string theory is studied. For the deformation parameter
$q$ a root of unity, in addition to the relation of such values of $q$ with the
rational conformal field theory, the Fock space of each oscillator mode in the
Fubini-Veneziano operator formulation becomes truncated. Thus, based on general
physical grounds, the resulting spin-(mass)$^2$ relation is expected to be
below the usual linear trajectory. For such specific values of $q$, we find
that the linear Regge trajectory turns into a square-root trajectory as the
mass increases.
