The structural synthesis of planetary gear trains (PGTs) is helpful for innovating transmission systems in machinery. A great deal of research has been devoted to the synthesis of one-degree-of-freedom (1-DOF) PGTs over the past half century. However, most synthesis methods are limited to PGTs with no more than eight links. Moreover, the synthesis results are not consistent with each other. Until now, the inconsistency of synthesis results is still unresolved and exact synthesis results remain elusive. This paper presents a systematic and fully automatic method based on parent graphs to synthesize 1-DOF PGTs. The complete database of rotation graphs (r-graphs) and displacement graphs (d-graphs) of 1-DOF PGTs with up to nine links is established for the first time. All possible reasons for the contradictory synthesis results in the literature are analyzed and the controversy in the existing synthesis results which has lasted for nearly half a century is completely resolved. The exact results of the 6-, 7-, and 8-link r-graphs are confirmed to be 27, 152, and 1070, respectively. The exact results of the 6-, 7-, and 8-link d-graphs are confirmed to be 81, 647, and 6360, respectively. Additionally, the new results of 8654 r-graphs and 71,837 d-graphs of 9-link PGTs are provided for the first time.