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 \(g\mapsto cg^{\prime }\) (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).

在本文中，我们研究了Dirichlet-to-Neumann映射，以求出树上均值公式的解。我们给出了Dirichlet-to-Neumann图的两个替代定义。对于第一个定义（涉及“梯度”与“法向矢量”的乘积）和有向树上的线性均值公式（仅考虑给定节点的后继），我们得出Dirichlet- to-Neumann映射由\（g \ mapsto cg ^ {\ prime} \）给出（此处c是一个显式常量）。请注意，这是一阶的本地运算符。我们还考虑线性无向均值公式（不仅考虑给定节点的继任者，还考虑祖先和继任者），并证明相似的结果。对于这种平均值公式，我们为相关的Dirichlet问题包括了一些存在性和唯一性结果。最后，我们给出了Dirichlet-to-Neumann映射的替代定义（考虑了树的给定分支上的差异）。使用此替代定义，对于一定范围的参数，我们获得了由非局部算符给出的Dirichlet-to-Neumann映射（就像在欧几里得空间中的经典拉普拉斯算子一样）。