A phylogenetic tree: both blue and red groups are monophyletic. The green group is paraphyletic: it is missing a monophyletic subgroup -- the blue group -- that shares a common ancestor with itself (that being the vertical stem at the root of the green group). In this form, monophyletic means "no sideways stems leaving the group".
A cladogram of the primates, showing a monophyletic taxon: the simians (in yellow); a paraphyletic taxon: the prosimians (in cyan, including the red patch); and a polyphyletic group: the night-active primates, i.e., the lorises and the tarsiers (in red)
A cladogram of the vertebrates showing phylogenetic groups. A monophyletictaxon (in yellow): the group of "reptiles and birds", contains its most recent common ancestor and all descendants of that ancestor. A paraphyletic taxon (in cyan): the group of reptiles, contains its most recent common ancestor, but does not contain all the descendants (namely Aves) of that ancestor. A polyphyletic "group" (in red): the group of all warm-blooded animals (Aves and Mammalia), does not contain the most recent common ancestor of all its members; this group is not seen as a taxonomic unit and is not considered a taxon by modern systematists.
In biological cladistics for the classification of organisms, monophyly is the condition of a taxonomic grouping being a clade — that is, a grouping of taxa which meets these criteria:
the grouping contains its own most recent common ancestor (or more precisely an ancestral population), i.e. excludes non-descendants of that common ancestor
the grouping contains all the descendants of that common ancestor, without exception
Monophyly is contrasted with paraphyly and polyphyly as shown in the second diagram. A paraphyletic grouping meets 1. but not 2., thus consisting of the descendants of a common ancestor excepting one or more monophyletic subgroups. A polyphyletic grouping meets neither criterion, and instead serves to characterize convergent relationships of biological features rather than genetic relationships -- for example, night-active primates, fruit trees, or aquatic insects. As such, these characteristic features of a polyphyletic grouping are not inherited from a common ancestor, but evolved independently.
Monophyletic groups are typically characterised by shared derived characteristics (synapomorphies), which distinguish organisms in the clade from other organisms. An equivalent term is holophyly.
The word "mono-phyly" means "one-tribe" in Greek.
These definitions have taken some time to be accepted. When the cladistics school of thought became mainstream in the 1960s, several alternative definitions were in use. Indeed, taxonomists sometimes used terms without defining them, leading to confusion in the early literature, a confusion which persists.
The first diagram shows a phylogenetic tree with two monophyletic groups. The several groups and subgroups are particularly situated as branches of the tree to indicate ordered lineal relationships between all the organisms shown. Further, any group may (or may not) be considered a taxon by modern systematics, depending upon the selection of its members in relation to their common ancestor(s); see second and third diagrams.
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^Hennig, Willi (1999) . Phylogenetic Systematics. Translated by Davis, D.; Zangerl, R. (Illinois Reissue ed.). Board of Trustees of the University of Illinois. pp. 72–77. ISBN978-0-252-06814-0.
^Aubert, D. 2015. A formal analysis of phylogenetic terminology: Towards a reconsideration of the current paradigm in systematics. Phytoneuron2015-66:1–54.