<p>A sofic-Dyck shift is a shift space presented by a Dyck automaton, which is a labelled graph equipped with a set of matching edges. As an example, a Dyck automaton can be constructed by selecting a specific vertex on a typical labelled graph, and attaching new loops with some matching brackets as their labels. We call the shift space presented by this particular automaton as a pointed sofic-Dyck shift. The complexity of this shift dynamical system can be understood through the distribution of its closed orbits. For this purpose, the prime orbit and Mertens’ orbit counting functions are used to denote the growth of the closed orbits in a certain way. In this paper, we will prove the asymptotic behaviours of the counting functions for pointed sofic-Dyck shifts via their Artin-Mazur zeta function. Using this approach, we investigate the analyticity of the zeta function and it leads to the desired result. This finding generalizes the established results on orbit growths of other types of shift spaces such as Dyck, Motzkin and bouquet-Dyck shifts.</p>