This paper studies the boundary behaviour of λ -polyharmonic functions for the simple random walk operator on a regular tree, where λ is complex and | λ | > ρ , the ℓ 2 -spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential Fatou theorem is proved.