When the identity map from the Dirichlet space on the polydisc to L2(μ) is bounded for a positive measure on the polydisc? This problem can be reduced to weighted Hardy inequality on a multi-tree. The weighted Hardy inequality on the usual tree can be reduced to the Carleson embedding theorem, a.k.a the boundedness of weighted dyadic paraproducts. But on graphs with cycles, e.g. on the direct product of two dyadic trees this inequality poses many combinatorial difficulties. We will explain how to cope with them on bi-tree and tri-tree and what this entails for a natural embedding of spaces of holomorphic functions on bi-disc and tri-disc. We finally obtain necessary and sufficient conditions for the embedding of the Dirichlet space into L2(μ) for b-disc and try-disc. Notice that even embedding theorem for a classical Hardy space on bidisc are not known when I am writing this abstract. This talk is based on series of joint works with N. Arcozzi, I. Holmes, P. Mozolyako, G. Psaromiligkos and P. Zorin-Kranich.