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Cladistics is the hierarchical classification of species based on evolutionary ancestry. In other words, it is the representation of the evoloution of species in a tree-like diagram, where each branch represents a different species. Each branch on an evolutionary tree is referred to as a "clade". A clade is a group of organisms that includes an ancestor and all descendents of that ancestor. You can think of a clade as a branch on the tree of life. http://evolution.berkeley.edu/evolibrary/article/phylogenetics_01
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Evolution works by species splitting. The "tree of life" consists of a huge number of two way splits where one species, for any one of many reasons, splits into two. Cladistics is just the attempt to discover this tree by finding which species split from which in what order. Each branch of any split is called a "clade", and for any three species, by definition two will be on one clade and one on the other (Though ther may be many other species on each clade). Cladistics is endevouring to find which are a pair by finding characteristics which two share and one does not.
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