Asymptotic normality of number of multiple coincidences of chains in complete $q$-ary trees and forests with randomly marked vertices

We consider complete $q$-ary trees of height $H$ with vertices marked by random independent marks taking values from some the set $\{1,2,\ldots, N\}$. The object of our research is the number of tuples of $r\ge 2$ paths having the same length $s$ and identical sequences of vertex marks. We propose a theorem on sufficient conditions of asymptotic normality of considered random values for the case when height of the tree tends to infinity. We also investigate repetitions of chains in forest of trees and suppose that there are $r$ trees which may have different heights $H_1, \ldots,H_r$ and vertices of these trees are marked in the same way. We consider the number of tuples of $r$ paths of the same length $s$ and suppose that a tuple includes one path from each tree. For such numbers of tuples, we also propose similar theorem on sufficient conditions of asymptotic normality.