On Harmonic Functions on Trees

By a tree T we mean a connected graph such that every subgraph obtained from T by removing any of its edges is not connected. In what follows we will only consider trees in which we distinguish a vertex v0 as an origin. As usual we denote by V and E the set of vertices and the set of edges (respectively) of the tree. If v and w are the boundary vertices of an edge, we say that they are neighbours and we write v ∼ w; we denote by [v,w] the edge that joins the vertices v and w. We assume (except for Sections 2 and 5) that the set of edges E is symmetric, i.e. [v,w] ∈ E if and only if [w, v] ∈ E. By a function on T we mean a function with real values defined on the set V of vertices of T and by a vector field we mean a function with real values defined on the set E of edges of T . If u is a function on T , its gradient ∇u is the vector field defined by the formula