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In cladistics, a synapomorphy or synapomorphic character is a trait that is shared ("symmorphy") by two or more taxa and their most recent common ancestor, whose own ancestor in turn does not possess the trait. A synapomorphy is thus an apomorphy visible in multiple taxa, where the trait in question originates in their last common ancestor. The word "synapomorphy" is derived from the Greek words σύν, syn = with, in company with, together with; ἀπό, apo = away from; and μορφή, morphe = shape.
True synapomorphies usually uniquely characterise a given set of terminal groups, but this is not essential to the concept. Thus, if some descendants of a last common ancestor possess a synapomorphic trait, it is not strictly necessary that all of its descendants must possess the same trait.
A synapomorphy should not be confused with other types of shared traits:
- A synapomorphy is a shared trait found among two or more taxa and their most recent common ancestor, whose ancestor in turn does not possess the trait. An example is the dipteran halteres, the uniquely modified hind wings found in all families of winged flies. No other group of insects possesses similar structures (in place of hind wings—insects in the order Strepsiptera have convergently-evolved halteres in place of fore wings). However, the fact that the trait is found exclusively in Diptera, to the exclusion of all other groups, is not essential in identifying the trait as a synapomorphy; rather, this fact makes its determination easier.
- A symplesiomorphy is a shared trait found among two or more taxa, but which is also found in taxa with an earlier common ancestor. An example of this is the five toes seen on the hind legs of rats and apes. This character-state originated very early in Tetrapoda and occurs in other tetrapod groups, e.g. in lizards. There is thus no indication that the group formed of rats and apes is a clade to the exclusion of these other groups.
- A homoplasy is a shared trait found among different taxa but not in their common ancestor (i.e., the same trait emerged in different taxa independently of each other). An example of this is homeothermy in birds and mammals. This trait is a derived character-state (in relation to poikilothermy, the character-state of the last common ancestor of both groups), which evolved independently in these two groups (or at least in the larger clades to which these groups belong).
Cladistic analyses 
Synapomorphies are used to establish phylogenies in cladistic analyses. As such they are empirical data which can support a certain hypothesis that terminal groups form a clade (monophyletic group) together to the exclusion of certain other groups – whereas character-states that are shared, but also shared by other terminal groups descending from an earlier common ancestor, cannot be used to exclude these other groups. The latter character-states can consist of symplesiomorphies ("primitive" character-states having originated in the earlier common ancestor) or homoplasies (superficially similar but independently evolved derived character-states).
The key problem is to identify the polarity of the transformation series to which several character-states belong, i.e. to tell which character-state is apomorphic and which is plesiomorphic. Various criteria were used to polarise the transformation series in earlier cladistics; however in the recent two decades pattern criteria based on outgroup comparison have dominated the field.
The concepts of apomorphy and plesiomorphy are relative to a certain level of generality. What counts as an apomorphy at one level of generality may well be a plesiomorphy at another. For example, for rats and apes, the presence of mammary glands is a symplesiomorphy, but it is a synapomorphy for mammals in relation to tetrapods more broadly.
It is not essential to a synapomorphy that all members of a clade possess it; even if some would have secondarily lost the trait it could still be a synapomorphy of the clade as a whole. A character state that is a synapomorphy for a clade, but for lineages in this clade is a plesiomorphy that is altered in some lineages, is called an underlying synapomorphy. If no crown group taxa are known, it is sometimes difficult to decide which character state is the underlying synapomorphy and which the autapomorphy that overlies it.
Clades are not defined by synapomorphies as such, though it is possible to define them by apomorphies in general.
Relative apparent synapomorphy analysis is a method used to determine whether a given character is shared between taxa due to shared ancestry or due to convergence.