Perfect phylogeny
Perfect phylogeny is a term used in computational phylogenetics to denote a phylogenetic tree in which all internal nodes may be labeled such that all characters evolve down the tree without homoplasy. That is, characteristics do not hold to evolutionary convergence, and do not have analogous structures. Statistically, this can be represented as an ancestor having state "0" in all characteristics where 0 represents a lack of that characteristic. Each of these characteristics changes from 0 to 1 exactly once and never reverts to state 0. It is rare that actual data adheres to the concept of perfect phylogeny.
In general there are two different data types that are used in the construction of a phylogenetic tree. In distance-based computations a phylogenetic tree is created by analyzing relationships among the distance between species and the edge lengths of a corresponding tree. Using a character-based approach employs character states across species as an input in an attempt to find the most "perfect" phylogenetic tree.
The statistical components of a perfect phylogenetic tree can best be described as follows:
A perfect phylogeny for an n x m character state matrix M is a rooted tree T with n leaves satisfying:
i. Each row of M labels exactly one leaf of T
ii. Each column of M labels exactly one edge of T
iii. Every interior edge of T is labeled by at least one column of M
It is worth noting that it is very rare to find actual phylogenetic data that adheres to the concepts and limitations detailed here. Therefore, it is often the case that researchers are forced to compromise by developing trees that simply try to minimize homoplasy, finding a maximum-cardinality set of compatible characters, or constructing phylogenies that match as closely as possible to the partitions implied by the characters.
Both of these data sets illustrate examples of character state matrices. Using matrix M'1 one is able to observe that the resulting phylogenetic tree can be created such that each of the characters label exactly one edge of the tree. In contrast, when observing matrix M'2, one can see that there is no way to set up the phylogenetic tree such that each character labels only one edge length.
An example of a character matrix that can be depicted as a perfect phylogeny
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