Fixed elements of infinite trees (Q1331970)

The automorphism group of a rayless tree fixes a vertex or a set of two adjacent vertices. The authors provide a short proof of this fact. For an alternate approach see [\textit{R. Schmidt}, Ein Ordnungsbegriff für Graphen ohne unendliche Wege mit einer Anwendung auf \(n\)-fach zusammenhängende Graphen, Arch. Math. 40, 283-288 (1983; Zbl 0526.05058)]. The same conclusion is shown to hold for trees which contain at least three ends of maximum order and no subgraph homeomorphic to the dyadic tree.