  In this paper we study the Dirichlet-to-Neumann map for solutions to mean
value formulas on trees. We give two alternative definition of the
Dirichlet-to-Neumann map. For the first definition (that involves the product
of a "gradient" with a "normal vector" and for a linear mean value formula on
the directed tree (taking into account only the successors of a given node) we
obtain that the Dirichlet-to-Neumann map is given by $g\mapsto cg'$ (here $c$
is an explicit constant). Notice that this is a local operator of order one. We
also consider linear undirected mean value formulas (taking into account not
only the successors but the ancestor and the successors of a given node) and
prove a similar result. For this kind of mean value formula we include some
existence and uniqueness results for the associated Dirichlet problem. Finally,
we give an alternative definition of the Dirichlet-to-Neumann map (taking into
account differences along a given branch of the tree). With this alternative
definition, for a certain range of parameters, we obtain that the
Dirichlet-to-Neumann map is given by a nonlocal operator (as happens for the
classical Laplacian in the Euclidean space).
