As a result of Charles Darwin’s and Alfred Russell Wallace’s discovery of evolution in the in the mid 19th Century, there has been interest in representing biologic taxonomic classifications as a tree-shaped diagram that represents true evolutionary pathways. This began shortly after evolution was recognized. In 1866 the German biologist Ernst Haeckel drew a tree of all life as the basis of a taxonomic classification (man at the apex). 
Dr Haeckel, of the University of Jena (1862 to 1909), was very influential in popularizing evolutionary theory. He discovered and named thousands of new species and developed the “ontogeny recapitulates phylogeny” theory. Some of this man’s more flamboyant ideas have been refuted, but his energy sparked a surge toward evolutionary taxonomy that is still gaining strength today. His notion of the rank of phylum implies a monophyletic source of life and that phylogeny should be used as a basis for taxonomy. The modern impetus of this concept is Willi Hennig, who originated the idea (but not the term) of cladistics. This hierarchial classification of species is based entirely on evolutionary ancestry which is demonstrated by a branching tree-like diagram known as cladograms (klados Gr. = branch). Hennig referred to this as Synapomorphy and the term cladistics was actually originated by Julian Huxley several years earlier, but the two terms have become essentially synonymous.
Phylogenetic /cladistic taxonomy contrasts with the classic Linnaean rank-based approach which classifies on the basis of morphologic similarity and, although this is logically distinct, there is some commonality in the result because evolution is a branching process whereby species separate into separate branches but retain morphologic similarities which then diminish as the evolutionary branching proceeds. The method of classifying on a quantitative morphologic (degree of neighboring) basis is called phenetics, numerical taxonomy or taximetrics. This is still philosophically distinct from phylogenetics, but of some of these methods have found their way into cladistics which has exploited more sophisticated methods of comparison such as Bayesian Inference, Maximum Likelihood and Parsimony. Cladistics became the taxonomic fashion of the late 20th Century and has continued into the 21st.
The cladistic approach has been enhanced by two concurrent developments: (a) the use of polymerase chain reactions (“PCR probes”) which determine base sequences in key portions of nuclear, chloroplast and ribosomal nucleic acids and (b) computers which can be programmed to manage great quantities of data in the production of the simplest, most efficient (“maximum parsimony”) branching diagrams. It must be said that there is always some element of assumption in these tool’s ability to produce the authentic phylogenetic tree.
The commonly used methods to infer phylogenetics include Bayesian inference, maximum likelihood and maximum parsimony.
(a) Bayesian Inference: The name “Bayesian” refers to Bayes’ Theorem developed by the reverend Thomas Bayes. In 1736, this Presbyterian minister and mathematician anonymously published a defense of the logical foundations of Isaac Newton’s calculus and was elected as a Fellow of the Royal Society in 1742. His solutions to probability interpretations have in common the idea of probability as a partial belief rather than a frequency and have been posthumously arranged to produce “Bayes” law or theorem: P(A/B) = P(B/A)P(A)/P(B) which determines conditional probablilty of A, depending on the specified value of B.
(b) Maximum Likelihood (MLE) is a statistical method for fitting a statistical model to data which provides the greatest likelihood of a population mean, based on a sample number and the variance of that sample. It is a method easily adapted to produce mean values and Gaussian normal distribution curves.
(c) Maximum Parsimony is a group of tree determination methods which design an optimally simple cladistic tree from sets of data pairs. This requires considerably more computer power than is in any human brain because there is such a huge number of possible trees. For example: 10 species with no predetermined root could have over two million possible trees!
The central concept in Cladistics is the clade—a group which includes the ancestor and all of its descendant species and no other species. This is also known as a monophyletic group. These words (clade, monophyletic) always are defined in terms of the ancestral starting point. If the name used to describe a group includes some species that belong in another monophyletic group, that (named) group is polyphyletic. If the named group contains the most but not all of the species, it is said to be paraphyletic.
(A) Monophyletic clades; (B) a paraphyletic clade; (C) a polyphyletic clade
An example of a polyphletic group would be ”warm blooded animals” which include two monophyletic groups —–birds (descendants of Archosaurs) and mammals (descendants of Eutheria). A paraphyletic group is exemplified by the name “reptiles” which, in common usage omits birds which share a common ancestor with the reptiles. Polyphyletic and paraphyletic groups have their origin in traditional taxonomy, based on similar morphological (“convergent evolution”) charactictics or in common parlance where there are functional similarities.
In Cladistics the monophyletic group, which strictly follows all of the evolutionary successors from a common ancestor, is the standard. Evolution is a branching process and the goal is to represent species relationships by the use of bifurcating (two way forks) tree-like diagrams which are known as cladograms. The ultimate (unattainable) cladogram would resemble the “tree of life” drawn by Ernst Haeckel in 1866. Is all life monophyletic? Do all animals, plants, eubacteria, archaea share a common ancestor? It’s doubtful, but probably we’ll never know. aeHaeckelHaeckel