  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 $d^2$ strictly positive random variables. For
any two connected vertices of the tree assign to the edge between them {\it 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.
{\it 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 $x\downarrow 0$, and apply the results to
some other relevant models.
