We obtain new results for the probabilistic model introduced in Menshikov et al. (2007) and Volkov (2006) which involves a d-ary regular tree. All vertices are coloured in one of d distinct colours so that d children of each vertex all have different colours. Fix d2 strictly positive random variables. For any two connected vertices of the tree assign to the edge between them a label which has the same distribution as one of these random variables, such that the distribution is determined solely by the colours of its endpoints. A value of a vertex is defined as a product of all labels on the path connecting the vertex to the root. We study how the total number of vertices with value of at least x grows as x[downwards arrow]0, and apply the results to some other relevant models.
