Maximum parsimony is a simple but popular technique used in cladistics to infer a phylogenetic tree for a set of taxa (commonly a set of species or reproductively-isolated populations of a single species) on the basis of some observed data on the similarities and differences among taxa.
The input data used in a maximum parsimony analysis is in the form of "characters" for a range of taxa. A character is a partitioning of the taxa into distinct character states with respect to some feature. A character could be binary (two state) e.g. for the presence or absence of a feature (e.g. tail), or it could be a multistate e.g. the protein or nucleic acid residue at a particular site in the organism's genome. Differences in character state are explained by evolutionary changes.
There are a number of probabilistic and deterministic algorithms available for constructing phylogenetic trees. However, what we describe here is the criterion that is used to define the best tree under maximum parsimony not the specific algorithm used to find that answer.
The trees used in maximum parsimony analysis are, in a general way, unrooted trees (there is no indication of time in the tree, only the relations between taxa). All the taxa used in the analysis are leaf nodes (often called tips or terminal taxa) in the (leaf-labelled) tree (so have only one edge into them). Internal nodes are inserted into the tree to represent the inferred ancestral species. Each internal node has at least three edges into it. Transitions between character states are associated with the edges on the tree.
Maximum parsimony trees are those that allow the explanation of the observed distribution of character states across taxa with the fewest inferred changes between character states. To do this, all trees are given a length, equal to the minimum number of transitions (changes between character states) which can explain the observed distribution of character states across taxa (leaves) assuming the tree. The trees with the shortest length are the maximum parsimony trees. This assumption that "the simplest possible explanation is the best" is considered by some to be a generalisation of Occam's Razor.
Problems with maximum parsimony
Maximum parsimony is a very simple approach, and is popular for this reason. However, it is not statistically consistent. That is, it is not guaranteed to produce the true tree with high probability, given sufficient data.
Although decreasingly of concern as geneotyping technology provides more characters, a major problem for paleontology with maximum parsimony is that it assumes that the only way two species can share the same character is if they are genetically related. This is not always the case. For example, birds and bats have wings, while crocodiles and humans do not. Based on this data, maximum parsimony would tend to group crocodiles with humans, and birds with bats. However, humans are actually more closely related to bats (a mammal) than crocodiles (a reptile) or birds. A similar problem occurs to back-mutation, where, due to a small mutation, a character is removed, but then it later comes back. Because maximum parsimony uses unrooted trees, independent innovation and back-mutation are actually mathematically equivalent.