Comparative Morphology
The first use of taxonomy was the hierarchical organization of plants and animals. Carl von Linne first devised a system for categorizing plant and animal life with the publication of the Systema Naturae in the 18th century. Subsequent generations of biologists finely honed this technique throughout the 19th century using phenemics, which compares animal morphology. Comparative morphology takes any visible or readily observable trait of an animal and uses similarities and differences in these traits to infer the common ancestry (or lack thereof) of different organisms. This technique is still used to supplement genetic studies, especially when such data is not readily available. For instance, paleontologists can easily distinguish between different groups of extinct trilobites based on their size, number of segments, and shape of the head and eyes.
Biochemical Taxonomy
Beginning in the 1980s, it became feasible to sequence long stretches of an animal's DNA. By comparing homologies (similarities) within large stretches of DNA, biologists can determine the relative relationships of many animals simultaneously. Similarities in protein structure and sequence are also sometimes used to infer homology (similarity). Scientists determine how biochemically similar different organisms are and build family trees based on those similarities. The similarity of these genetic trees with phenetic trees devised by previous generations of scientists is a powerful proof for the theory of evolution. Furthermore, these biochemical similarities are consistent within stretches of DNA, sequences of protein, or the arrangement of genes within an organism's genome.
Non-Scientific Taxonomies
Any phenomenon that expresses both diversity and inheritance is potentially classifiable by taxonomy. For instance, linguists employ mathematical techniques such as principal component analysis to compare the grammatical structure and phonemes of different languages to build language trees. Spanish, Portuguese, French, and Italian are all "Romance" languages derived from Latin. Spanish and Portuguese are both Iberian Romance languages derived from a common ancestor language. Technologies can also be explored taxonomically. For instance, new generations of computer technology are built upon similar underlying technologies, but different devices diverge and specialize over time to more effectively and efficiently approach computing problems. Most personal computers are built on an x86 microchip architecture. These chips are given taxonomic designations based on their manufacturer, transistor size, and speed.
Computational Taxonomy
Even with modern biochemical techniques, taxonomic trees are fairly crude without complex mathematical models to confirm and calculate confidence for different arrangements of a relational "family tree" of related items. These computational models use advanced systems such as Bayesian trees and random forests to calculate the relative fitness of different relational trees. The random forest model has proven especially effective at calculating these relationships. In this technique, many individual branching "trees" using one or several measures of comparison are randomly generated. These individual trees are then compared en masse. The relational tree with the greatest mixture of simplicity and predictive power is output from this model. Such advanced computing techniques can effectively compute taxonomic trees using small sample sizes with many measurable traits.