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Property of a group which includes only descendants of a common ancestor, but excludes at least one monophyletic subgroup
In this phylogenetic tree, the green group is paraphyletic; it is composed of a common ancestor (the lowest green vertical stem) and its descendants, but it excludes the blue group (a monophyletic group) which diverged from the green group.
In taxonomy, a group is paraphyletic if it consists of the group's last common ancestor and all descendants of that ancestor excluding a few--typically only one or two--monophyletic subgroups. The group is said to be paraphyletic with respect to the excluded subgroups. A paraphyletic group cannot be a clade, or monophyletic group, which is any group of species that includes a common ancestor and all of its descendants. One or more members of a paraphyletic group is more closely related to the excluded group(s) than it is to other members of the paraphyletic group. The term is commonly used in phylogenetics (a subfield of biology) and in linguistics. Paraphyletic groups are identified by a combination of synapomorphies and symplesiomorphies.
The term was coined by Willi Hennig to apply to well-known taxa like Reptilia (reptiles) which, as commonly named and traditionally defined, is paraphyletic with respect to mammals and birds. Reptilia contains the last common ancestor of reptiles and all descendants of that ancestor, including all extant reptiles as well as the extinct synapsids, except for mammals and birds. Other commonly recognized paraphyletic groups include fish, monkeys, and lizards.[page needed]
If many subgroups are missing from the named group, it is said to be polyparaphyletic.
The term paraphyly, or paraphyletic, derives from the two Ancient Greek words ? (pará), meaning "beside, near", and (phûlon), meaning "genus, species", and refers to the situation in which one or several monophyletic subgroups of organisms (e.g., genera, species) are left apart from all other descendants of a unique common ancestor.
Conversely, the term monophyly, or monophyletic, builds on the Ancient Greek prefix (mónos), meaning "alone, only, unique", and refers to the fact that a monophyletic group includes organisms consisting of all the descendants of a unique common ancestor.
By comparison, the term polyphyly, or polyphyletic, uses the Ancient Greek prefix (polús), meaning "many, a lot of", and refers to the fact that a polyphyletic group includes organisms arising from multiple ancestral sources.
Reptilia (green field) is a paraphyletic group comprising all amniotes (Amniota) except for two subgroups: Mammalia (mammals) and Aves (birds); therefore, Reptilia is not a clade. In contrast, Amniota itself is a clade, which is a monophyletic group.
Groups that include all the descendants of a common ancestor are said to be monophyletic. A paraphyletic group is a monophyletic group from which one or more subsidiary clades (monophyletic groups) are excluded to form a separate group. Philosopher of science Marc Ereshefsky has argued that paraphyletic taxa are the result of anagenesis in the excluded group or groups. Cladists do not grant paraphyletic assemblages the status of "groups" or reify them with explanations, because they represent evolutionary non-events 
A group whose identifying features evolved convergently in two or more lineages is polyphyletic (Greek [polys], "many"). More broadly, any taxon that is not paraphyletic or monophyletic can be called polyphyletic. Empirically, the distinction between polyphyletic groups and paraphyletic groups is rather arbitrary, since the character states of common ancestors are inferences, not observations.
These terms were developed during the debates of the 1960s and 1970s accompanying the rise of cladistics.
Paraphyletic groupings are considered problematic by many taxonomists, as it is not possible to talk precisely about their phylogenetic relationships, their characteristic traits and literal extinction. Related terms are stem group, chronospecies, budding cladogenesis, anagenesis, or 'grade' groupings. Paraphyletic groups are often relics from outdated hypotheses of phylogenic relationships from before the rise of cladistics.
Wasps are paraphyletic, consisting of the cladeApocrita without ants and bees, which are not usually considered to be wasps; the sawflies ("Symphyta") too are paraphyletic, as the Apocrita are nested inside the Symphytan clades.
The prokaryotes (single-celled life forms without cell nuclei), because they exclude the eukaryotes, a descendant group. Bacteria and Archaea are prokaryotes, but archaea and eukaryotes share a common ancestor that is not ancestral to the bacteria. The prokaryote/eukaryote distinction was proposed by Edouard Chatton in 1937 and was generally accepted after being adopted by Roger Stanier and C.B. van Niel in 1962. The botanical code (the ICBN, now the ICN) abandoned consideration of bacterial nomenclature in 1975; currently, prokaryotic nomenclature is regulated under the ICNB with a starting date of 1 January 1980 (in contrast to a 1753 start date under the ICBN/ICN).
Among plants, dicotyledons (in the traditional sense) are paraphyletic because the group excludes monocotyledons. "Dicotyledon" has not been used as a botanic classification for decades, but is allowed as a synonym of Magnoliopsida.[note 1] Phylogenetic analysis indicates that the monocots are a development from a dicot ancestor. Excluding monocots from the dicots makes the latter a paraphyletic group.
Among animals, several familiar groups are not, in fact, clades. The order Artiodactyla (even-toed ungulates) as traditionally defined is paraphyletic because it excludes Cetaceans (whales, dolphins, etc.). Under the ranks of the ICZN Code, the two taxa are separate orders. Molecular studies, however, have shown that the Cetacea descend from artiodactyl ancestors, although the precise phylogeny within the order remains uncertain. Without the Cetaceans the Artiodactyls are paraphyletic.
The class Reptilia, as traditionally defined, is paraphyletic because it excludes birds (class Aves) and mammals. Under the ranks of the ICZN Code, these three taxa are separate classes. However, mammals hail from the synapsids (which were once described as "mammal-like reptiles") and birds are sister taxon to a group of dinosaurs (part of Diapsida), both of which are "reptiles". Alternatively, reptiles are paraphyletic because they gave rise to (only) birds. Birds and reptiles together make Sauropsids, a clade of Amniota that is the sister group of the clade that includes mammals.
One of the goals of modern taxonomy over the past fifty years has been to eliminate paraphyletic "groups" such as the examples here from formal classifications. 
Paraphyly in species
Species have a special status in systematics as being an observable feature of nature itself and as the basic unit of classification. Some articulations of the phylogenetic species concept require species to be monophyletic, but paraphyletic species are common in nature, to the extent that they do not have a single common ancestor. Indeed, for sexually reproducing taxa, no species has a "single common ancestor" organism. Paraphyly is common in speciation, whereby a mother species (a paraspecies) gives rise to a daughter species without itself becoming extinct. Research indicates as many as 20 percent of all animal species and between 20 and 50 percent of plant species are paraphyletic. Accounting for these facts, some taxonomists argue that paraphyly is a trait of nature that should be acknowledged at higher taxonomic levels.
Cladists advocate a phylogenetic species concept  that does not consider species to exhibit the properties of monophyly or paraphyly, concepts under that perspective which apply only to groups of species.  They consider Zander's extension of the "paraphyletic species" argument to higher taxa to represent a category error
Uses for paraphyletic groups
When the appearance of significant traits has led a subclade on an evolutionary path very divergent from that of a more inclusive clade, it often makes sense to study the paraphyletic group that remains without considering the larger clade. For example, the Neogene evolution of the Artiodactyla (even-toed ungulates, like deer, cows, pigs and hippopotamuses - note that Cervidae, Bovidae, Suidae and Hippopotamidae, the families that contain these various artiodactyls, are all monophyletic groups) has taken place in environments so different from that of the Cetacea (whales, dolphins, and porpoises) that the Artiodactyla are often studied in isolation even though the cetaceans are a descendant group. The prokaryote group is another example; it is paraphyletic because it is composed of two Domains (Eubacteria and Archaea) and excludes (the eukaryotes). It is very useful because it has a clearly defined and significant distinction (absence of a cell nucleus, a plesiomorphy) from its excluded descendants.
Also, some systematists recognize paraphyletic groups as being involved in evolutionary transitions, the development of the first tetrapods from their ancestors for example. Any name given to these hypothetical ancestors to distinguish them from tetrapods--"fish", for example--necessarily picks out a paraphyletic group, because the descendant tetrapods are not included. Other systematists consider reification of paraphyletic groups to obscure inferred patterns of evolutionary history.
Current phylogenetic hypotheses of tetrapod relationships imply that viviparity, the production of offspring without the external laying of a fertilized egg, developed independently in the lineages that led to humans (Homo sapiens) and southern water skinks (Eulampus tympanum, a kind of lizard). Put another way, viviparity is a synapomorphy for Theria within mammals, and an autapomorphy for Eulamprus tympanum (or perhaps a synapomorphy, if other Eulamprus species are also viviparous).
"Groups" based on independently-developed traits such as these examples of viviparity represent examples of polyphyly, not paraphyly.
Flightless birds are polyphyletic because they independently (in parallel) lost the ability to fly.
Animals with a dorsal fin are not paraphyletic, even though their last common ancestor may have had such a fin, because the Mesozoic ancestors of porpoises did not have such a fin, whereas pre-Mesozoic fish did have one.
Quadrupedalarchosaurs are not a paraphyletic group. Bipedal dinosaurs like Eoraptor, ancestral to quadrupedal ones, were descendants of the last common ancestor of quadrupedal dinosaurs and other quadrupedal archosaurs like the crocodilians.
Non-exhaustive list of paraphyletic groups
The following list recapitulates a number of paraphyletic groups proposed in the literature, and provides the corresponding monophyletic taxa.
^Stackebrabdt, E.; Tindell, B.; Ludwig, W.; Goodfellow, M. (1999). "Prokaryotic Diversity and Systematics". In Lengeler, Joseph W.; Drews, Gerhart; Schlegel, Hans Günter (eds.). Biology of the prokaryotes. Stuttgart: Georg Thieme Verlag. p. 679.
^ abSimpson 2006, pp. 139-140. "It is now thought that the possession of two cotyledons is an ancestral feature for the taxa of the flowering plants and not an apomorphy for any group within. The 'dicots' ... are paraphyletic ...."
^O'Leary, Maureen A. (2001). "The phylogenetic position of cetaceans: further combined data analyses, comparisons with the stratigraphic record and a discussion of character optimization". American Zoologist. 41 (3): 487-506. CiteSeerX10.1.1.555.8631. doi:10.1093/icb/41.3.487.
^Romer, A. S. & Parsons, T. S. (1985): The Vertebrate Body. (6th ed.) Saunders, Philadelphia.
^ abcSharkey, M. J. (2007). "Phylogeny and classification of Hymenoptera"(PDF). Zootaxa. 1668: 521-548. doi:10.11646/zootaxa.1668.1.25. Symphyta and Apocrita have long been considered as suborders of Hymenoptera but since recognition of the paraphyletic nature of the Symphyta (Köningsmann 1977, Rasnitsyn 1988) and the advent of cladistic methods the subordinal classification should be avoided. Likewise the woodwasps are thought to be non-monophyletic, forming a grade that is ancestral relative to Apocrita and Orussidae. The traditional hymenopteran classification is faulty, by cladistic criteria,in the same way as pre-cladistic vertebrate classifications in which groups sharing plesiomorphic characterswere recognized as natural, e.g., fishes were once grouped together as 'Pisces', which excluded tetrapods.
^Parhi, J.; Tripathy, P.S.; Priyadarshi, H.; Mandal S.C.; Pandey P.K. (2019). "Diagnosis of mitogenome for robust phylogeny: A case of Cypriniformes fish group". Gene. 713: 143967. doi:10.1016/j.gene.2019.143967. PMID31279710.
^New data on Kimberella, the Vendian mollusc-like organism (White sea region, Russia): palaeoecological and evolutionary implications (2007), "Fedonkin, M.A.; Simonetta, A; Ivantsov, A.Y.", in Vickers-Rich, Patricia; Komarower, Patricia (eds.), The Rise and Fall of the Ediacaran Biota, Special publications, 286, London: Geological Society, pp. 157-179, doi:10.1144/SP286.12, ISBN9781862392335, OCLC156823511CS1 maint: uses authors parameter (link)
^Martindale, Mark; Finnerty, J.R.; Henry, J.Q. (September 2002). "The Radiata and the evolutionary origins of the bilaterian body plan". Molecular Phylogenetics and Evolution. 24 (3): 358-365. doi:10.1016/s1055-7903(02)00208-7. PMID12220977.
^Webster, Bonnie L.; Copley, Richard R.; Jenner, Ronald A.; Mackenzie-Dodds, Jacqueline A.; Bourlat, Sarah J.; Rota-Stabelli, Omar; Littlewood, D. T. J.; Telford, Maximilian J. (November 2006). "Mitogenomics and phylogenomics reveal priapulid worms as extant models of the ancestral Ecdysozoan". Evolution & Development. 8 (6): 502-510. doi:10.1111/j.1525-142X.2006.00123.x. PMID17073934. S2CID22823313.
^Greenhill, Simon J. and Russell D. Gray. (2009.) "Austronesian Language and Phylogenies: Myths and Misconceptions About Bayesian Computational Methods," in Austronesian Historical Linguistics and Culture History: a Festschrift for Robert Blust, edited by Alexander Adelaar and Andrew Pawley. Canberra: Pacific Linguistics, Research School of Pacific and Asian Studies, The Australian National University.
Simpson, Michael George (2006). Plant systematics. Burlington; San Diego; London: Elsevier Academic Press. ISBN978-0-12-644460-5.