The author gives a combinatorial proof of the well known theorem of Bass and \textit{J.-P. Serre} [Arbres, amalgams, \(SL_ 2\), Astérisque 46 (1977; Zbl 0369.20013)] on the structure of groups acting on trees. His proof is based on a theorem of \textit{A. M. Macbeath} [Ann. Math., II. Ser. 79, 473-488 (1964; Zbl 0122.175)] which gives a presentation for a group acting on a topological space in terms of translates of a fundamental region. A proof of Macbeath's theorem in the special case needed is included. The English translation contains a few mistakes, e.g. it should read ''Tietze transformations'' instead of ''Tits transf.''.