Cladists label Odontognathae as "paraphyletic", because cladistics does not recognize direct ancestral relationships.
In traditional classification, the Neornithes also included a third superorder, the Odontognathae, containing advanced toothed birds from the Cretaceous, like Hesperornis and Ichthyornis. This superorder is likely paraphyletic [ANCESTRAL], and falls outside [IS ANCESTRAL TO] the crown group birds. It is not entirely clear whether the Palaeognathae too are paraphyletic, or represent a primitve grade of birds.
These birds [Hesperornithes] were originally combined with Ichthyornis in the paraphyletic "Odontornithes" by Othniel Charles Marsh, in 1873. In 1875, they were separated as Odontolcae. The group was often considered to be allied to loons and grebes,or to the Paleognathae. These similarities, however, as the more recently determined fact that the osteons of their bones - at least in Hesperornis - were arranged in a pattern similar to that in Neognathae, are today considered [incorrectly] to be due to convergent evolution.
Odontornithes is an obsolete and disused taxonomic term proposed by O. C. Marsh for birds possessing teeth, notably the genera Hesperornis and Ichthyornis from the Cretaceous deposits of Kansas.In 1875 March divided this "subclass" into Odontolcae, with the teeth standing in grooves, and Odontotormae, with the teeth in separate alveoles or sockets. In his 1880 work, Odontornithes: A monograph on the extinct toothed birds of North America, he added the Saururae, represented by Archaeopteryx, as a third order.
The resulting classification was paraphyletic, [ANCESTRAL]
not accurately resolving evolutionary relationships, andso it has been abandoned by most modern scientists, though at least one 21st century paper re-used the concept under the older name Odontoholomorphae (first coined by Stejneger, 1885).
The best known of the "Odontornithes" are Hesperornis regalis, standing about 3 ft. high, the somewhat taller H. crassipes, and Ichthyornis dispar. Hesperornis looked somewhat similar to a loon, while Ichthyornis was quite similar to a gull or petrel.
 Livezey, B.C. & Zusi, R.L. (2007): Higher-order phylogeny of modern birds (Theropoda, Aves: Neornithes) based on comparative anatomy. II. Analysis and discussion. Zoological Journal of the Linnean Society no 149(1), pp 1-95
"A cladistic analysis of the skeletons of loons (Gaviidae), grebes (Podicipedidae), and the Cretaceous diving birds, Hesperornis and Baptornis, supports the hypothesis that they form a monophyletic group (here called the Gaviomorphae) within the class Aves. Two lineages within the gaviomorphs can be delineated: (1) loons + grebes, and (2) Hesperornis + Baptornis. The Early Cretaceous Enaliornis and the Late Cretaceous Neogaeornis are related to the second lineage; the Late Cretaceous Lonchodytes, often placed near loons, is apparently not a gaviomorph but perhaps a charadriiform. Skeletal evidence also suggests that penguins (Spheniscidae) are the sister-group of the Gaviomorphae. Arguments that similarities of gaviomorph taxa represent convergence are not well founded. First, morphological differences among taxa do not constitute valid evidence against their monophyly, as many previous workers have argued. Second, in no case has anyone supporting convergence presented a corroborated alternative phylogenetic hypothesis. A close relationship among gaviomorphs, penguins, and apparently also the Procellariiformes and Pelecaniformes implies that a lineage of aquatic birds was established very early in avian history, presumably in the Early Cretaceous or Late Jurassic." (Joel Cracraft 1982)
Odontognathae is a disused name for a paraphyletic [ancestral] group of toothed prehistoric birds. The group was originally proposed by Alexander Wetmore, who attempted to link fossil birds with the presence of teeth, specifically of the orders Hesperornithiformes and Ichthyornithiformes. As such they would be regarded as transitional fossils between the reptile-like "Archaeornithes" like Archaeopteryx and modern birds. They were described by Romer as birds with essentially modern anatomy, but retaining teeth.Unlike the dinosaur-like "Archaeornithes", the various types of birds assigned to the Odontognathae had short tails with a plowshare-shaped pygostyle and a well developed carina for flight muscle. They also shared the feature of intramandibular articulation, something that is actually absent in Archaeopteryx, but found in many of its theropod relatives.The brains of the "odontognath" birds appear to be somewhat simpler than those of modern birds and have retained some "reptilian" traits. Ornithologist Alan Feduccia has used this, and the presence of the intramandibular articulation (a trait also found in mosasaurs and living varanid lizards) as arguments that the Odontognathae and thus the birds as a whole have not evolved from theropod dinosaurs, but non-dinosaur thecodonts. This theory is contested by most paleontologists.Feduccia:
It would not tax the imagination to engender a long list of obstacles for the now dominant model of a theropod origin of birds, including....the sliding lower jaw joint [intramandibular joint] of theropods (absent in birds)
In traditional classification, it [Archaeornithes] is one of two subclasses of birds, the other subclass being the Neornithes, the birds with a short, modern tail. This classification was erected by Hans Friedrich Gadow in 1893 and followed by Alfred Romer(1933) and subsequent authors through most of the 20th century. Other mesozoic birds like the toothed, but otherwise modern, birds like Hesperornis were included under the latter in their own superorder, the Odontognathae.