Dynamical structure of metric and linear self-maps on combinatorial trees

The dynamical structure of metric and linear self-maps on combinatorial trees is described. Specifically, the following question is addressed: given a map from a finite set to itself, under what conditions there exists a tree on this set such that the given map is either a metric or a linear map on this tree? The author proves that a necessary and sufficient condition for this is that the map has either a fixed point or a periodic point with period two, in which case all its periodic points must have even periods. The dynamical structure of tree automorphisms and endomorphisms is also described in a similar manner.