  We prove multi-parameter dyadic embedding theorem for Hardy operator on the
multi-tree. We also show that for a large class of Dirichlet spaces in bi-disc
and tri-disc this proves the embedding theorem of those Dirichlet spaces of
holomorphic function on bi- and tri-disc. We completely describe the Carleson
measures for such embeddings. The result below generalizes embedding result of
\cite{AMPVZ} from bi-tree to tri-tree. One of our embedding description is
similar to Carleson--Chang--Fefferman condition and involves dyadic open sets.
On the other hand, the unusual feature of \cite{AMPVZ} was that embedding on
bi-tree turned out to be equivalent to one box Carleson condition. This is in
striking difference to works of Chang--Fefferman and well known Carleson quilt
counterexample. We prove here the same unexpected result for the tri-tree.
Finally, we explain the obstacle that prevents us from proving our results on
polydiscs of dimension four and higher.
