  This paper studies the boundary behaviour of $\lambda$-polyharmonic functions
for the simple random walk operator on a regular tree, where $\lambda$ is
complex and $|\lambda|> \rho$, the $\ell^2$-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.
