Plesiomorphy, symplesiomorphy, apomorphy, and synapomorphy, all mean a trait shared between species because they share an ancestral species.
Apomorphic and synapomorphic characteristics convey much information about evolutionary clades and can be used to define taxa. However, plesiomorphic and symplesiomorphic characteristics cannot.
A backbone is a plesiomorphic trait shared by birds and mammals, and does not help in placing an animal in one or the other of these two clades. Birds and mammals share this trait because both clades are descended from the same far distant ancestor. Other clades, e.g. snakes, lizards, turtles, fish, frogs, all have backbones and none are either birds nor mammals.
Being a hexapod is plesiomorphic trait shared by ants and beetles, and does not help in placing an animal in one or the other of these two clades. Ants and beetles share this trait because both clades are descended from the same far distant ancestor. Other clades, e.g. bugs, flies, bees, aphids, and many more clades, all are hexapods and none are either ants nor beetles.
Feathers are a synapomorphy for placing any living species into the bird clade, hair is a synapomorphy for placing any living species into the mammal clade, elytra are a synapomorphy for placing any living species into the beetle clade, and the metapleural gland is a synapomorphy for placing any living species into the ant clade.
Note that some mammal species have lost their hair, so the absence of hair does not exclude a species from being a mammal. Another mammalian synapomorphy is milk. All mammals produce milk and no other clade contains animals which produce milk. Feathers, and milk are also apomorphies.
All of these terms are by definition relative, in that a trait can be a plesiomorphy in one context and an apomorphy in another. E.g. having a backbone is plesiomorphic between birds and mammals, but is apomorphic between them and insects. That is birds and mammals are vertebrates for which the backbone is a defining synapomorphic characteristic, while insects are invertebrates for which the absence of a backbone is a defining characteristic.
Species should not be grouped purely by morphologic or genetic similarity. Because a plesiomorphic character inherited from a common ancestor can appear anywhere in a phylogenetic tree, its presence does not reveal anything about the relationships within the tree. Thus grouping species requires distinguishing ancestral from derived character states.
An example is thermo-regulation in Sauropsida which is the clade containing the lizards, turtles, crocodiles, and birds. Lizards, trutles, and crocodiles are ectothermic (coldblooded), while birds are endothermic (warmblooded). Being coldblooded is symplesiomorphic for lizards, turtles, and crocodiles, but they do not form a clade, as crocodiles are more related to birds than to lizards and turtles. Thus using coldbloddness as an apomorphic trait to group crocodiles with lizards, and turtles, would be an error, and thus it is a plesiomorphic trait shared by these three clades due to their distant common ancestry.
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