We consider the problem of characterizing the Carleson measures for the Dirichlet space on the bidisc and reduce it to a problem concerning a bilinear Hardy operator on the direct product of two trees, which can be solved. After giving a brief introduction to the Dirichlet and Hardy spaces of analytic functions, we introduce the basics of (logarithmic) potential theory on the bitree and investigate some naturally arising capacitary-type inequalities. Possible further inquiries and related problems are discussed. Work in collaboration with Nicola Arcozzi, Karl-Mikael Perfekt, and Giulia Sarfatti.