The recently proposed hypothesis that the living birds and the living mammals are sister groups, together forming a natural taxon Haemothermia, is contrasted with the more traditional view, that birds and crocodiles are living sister groups within the taxon Archosauria. Of the 28 or so characters claimed to be unique to the Haemothermia, several are found to be structurally or developmentally different in birds compared to mammals. Others are found to occur also in crocodiles. In either case the status of the characters as homologues categorizing the group becomes doubtful, and only about eight characters remain as potentially acceptable. In contrast, some 24 characters are identified as potential homologues of birds plus crocodiles and therefore categorizing the group Archosauria, and this hypothesis must be judged preferable. The evidence of the characters of fossils can be used legitimately to test cladograms, but only to a limited extent. Nevertheless, the relevant fossils do support the Archosauria rather than the Haemothermia hypothesis. Cladograms logically may, and methodologically should, be taken as theories of phylogenetic relationships, and are potentially subject to independent evolutionary tests. A priori character weighting is sound in principle but cannot be applied in practice for want of the necessary, independently acquired knowledge of how characters change in evolution. The relative dates of the fossils is shown to be more compatible with the Archosauria than the Haemothermia classification. Finally, the hypothetical common ancestors that are implied by the two respective cladograms are compared. That shared by mammals and birds, as implied by the Haemothermia theory, would have been functionally incongruent, and therefore less probable than that shared by birds and crocodiles. These several lines of evidence all lead to the conclusion that the traditional theory of a relationship between birds and crocodiles, vis à vis mammals is substantially the better supported.