On the asymptotic normality in the problem on the tuples repetitions in a marked complete tree

We consider complete $q$-ary trees of height $H$ with vertices marked by random independent marks taking values from some finite set. The number of pairs of paths having length $s$ and identical sequences of vertex marks is investigated. For the distribution of this number we propose sufficient conditions of asymptotic normality for the case when $H\to\infty$ and parameters $s$ and $q$ may vary.