  In this paper we study the Dirichlet problem for systems of mean value
equations on a regular tree. We deal both with the directed case (the equations
verified by the components of the system at a node in the tree only involve
values of the unknowns at the successors of the node in the tree) and the
undirected case (now the equations also involve the predecessor in the tree).
We find necessary and sufficient conditions on the coefficients in order to
have existence and uniqueness of solutions for continuous boundary data. In a
particular case, we also include an interpretation of such solutions as a limit
of value functions of suitable two-players zero-sum games.
