Hardy operator on the poly-tree and some unexpected combinatorial properties of planar measures

We consider embeddings of various spaces of analytic functions on the polydisc into Lebesgue spaces with respect to a measure on the polydisc. These problems can often be moved to a discrete setting by considering weighted Dirichlet spaces on (poly-)trees and weighted dyadic multi-parameter Hardy operators. We find necessary and sufficient conditions for this operator to be bounded in the $n$-parameter case, when $n$ is 1, 2, or 3. The answer is quite unexpected – it is a certain combinatorial property of all measures in dimension 2 and 3 — and seemingly goes against the well known difference between box and Chang–Fefferman condition that was given by Carleson quilts counterexample of 1974.