Conditions for asymptotic normality of the number of multiple repetitions of chains in marked complete trees and forests

We consider complete $q$-ary trees of height $H$ with vertices marked by random independent marks taking values from the set $\{1,2,\ldots, N\}$ and forests of such trees. For both cases we investigate the number of sets of $r\ge 2$ paths with fixed length $s$ such that corresponding $s$-chains of marks of vertices are identical. We propose three theorems on sufficient conditions for the asymptotic normality for considered random values as $H\to\infty$ and possibly varying parameters $s$ and $q$.