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Let all vertices of a complete  q -ary tree of finite height be independently and equiprobably labeled by the elements of some finite alphabet. We consider the numbers of pairs of identical tuples of labels on chains of subsequent vertices in the tree. Exact formulae for the expectations of these numbers are obtained, convergence to the compound Poisson distribution is proved. For the size of cluster composed by pairs of identically labeled chains we also obtain exact formula for the expectation.

On coincidences of tuples in a q -ary tree with random labels of vertices