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Crocodilia - Crocodilians
Rhynchocephalia - Tuataras
Squamata
Suborder Sauria - Lizards
Suborder Serpentes - Snakes
Testudines - Turtles
Superorder Dinosauria
Saurischia
Ornithischia
Reptiles classically included all the amniotes except birds and mammals. Thus reptiles were defined as the set of animals that includes crocodiles, alligators, tuataras, lizards, snakes, and turtles, grouped together as the class Reptilia. This is still the usual definition of the term.
However, in recent years many taxonomists have begun to insist that taxa should be monophyletic, that is, groups should include all descendants of a particular form. The reptiles as defined above would be paraphyletic, since they exclude both birds and mammals, although these also developed from the original reptile. Colin Tudge writes:
Some cladists thus redefine Reptilia as a monophyletic group, including both the classic reptiles as well as the birds and perhaps the mammals (depending on ideas about their relationships). Others abandon it as a formal taxon altogether, dividing it into several different classes. However, other biologists believe that the common characters of the standard four orders are more important than the exact relationships, or feel that redefining the Reptilia to include birds and mammals would be a confusing break with tradition. A number of biologists have adopted a compromise system, marking paraphyletic groups with an asterisk, e.g. class Reptilia*. Colin Tudge notes other uses of this compromise system:
Several thousand fossil species showing a clear smooth transition from the ancestors of reptiles to present-day reptiles exist.
The first true "reptile" or Amniotes are categoized as Anapsids, having a solid skull with holes only for nose, eyes, spinal cord, etc. Turtles are believed by some to be surviving Anapsids, as they also share this skull structure, but this point has become contentious lately, with some arguing that turtles reverted to this primitive state in order to improve their armor. Both sides have strong evidence, and the conflict has yet to be resolved.
Shortly after the first reptiles, two branches split off. One group, the Synapsida, had a pair of holes in their skull behind the eyes, which was used to both lighten the skull and to increase the space for jaw muscles. The other group, Diapsida, possesed the same holes, along with a second pair located higher on the skull. The Synapsida eventually evolved into mammals, while Diapsida split yet again into two lineages, the lepidosaurs (which contain modern snakes, lizards and tuataras, as well as (debatably) the extinct sea reptiles of the mesozoic) and the archosaurs (modernly represented by only crocodiles and birds, but containing pterosaurs and dinosaurs).