The wide application of quantitative taxonomy has promoted the rapid development of its mathematical theory. Various mathematical tools including set theory, graph theory, probability theory, statistics, linear algebra, information theory and fuzzy mathematics are introduced, and various mathematical tools have produced different classification methods. The diversity of methods meets the needs of various applications. The application of graph theory in combinatorial mathematics to classification produces graph theory classification. Directed tree graph and weighted directed tree graph in graph theory are important tools to describe the evolutionary relationship of organisms by mathematical methods, so graph theory becomes the mathematical theoretical basis of quantitative branching classification. Graph theory research on branch classification has developed rapidly. With the development of mathematical theory, a more perfect branch classification method is derived. Another development direction of graph theory method is to use the minimum spanning tree in weighted undirected graph theory to represent the biological classification relationship. It can also be used to characterize classification.

Graph theory classification, as a tool of branch classification, is widely used in the study of paleontology, heredity and evolution theory. Since 1970s, with the development of molecular biology, graph theory classification has been applied to the classification of protein, nucleic acids and other biological macromolecules. Describe the law of inheritance and evolution at the molecular level. The classification of graph theory is not perfect, and there are still many problems to be solved, but it develops rapidly because of its importance. Principal component analysis in multivariate statistics is also applied to classification. On the issue of classification, many characters are related. If a direction can be found in the multi-dimensional space with character coordinates, the deviation reflected by characters in this direction changes the most, and determining a vector along this direction is called first principal component, followed by the second, third, ... principal components. This method is like grasping the principal contradiction from a complex thing with multiple characteristics, and simplifying a multi-dimensional classification problem into a simple problem in a low-dimensional space by using the operation skills of grasping the principal contradiction. Therefore, the classification problem is easy to solve.

Principal component classification developed earlier, and its mathematical basis is the content of eigenvalues and eigenvectors in linear algebra. It is a very effective tool to understand complex biological problems. Fig. 4 shows the results of principal component analysis of China pig breeds. Through analysis, it is pointed out that the three main aspects of understanding China pig breeds are pig body shape, fecundity and meat fat yield. In the coordinate map corresponding to the three principal components, the positions of 48 main pig breeds are drawn, and the spatial description of China pig breeds is given very vividly. Principal component classification not only classifies organisms, but also is widely used in practical problems such as ecology, environmental science, agriculture, forestry and medicine. The classification method based on the concept of fuzzy set is called fuzzy classification. Because many biological performance traits belong to fuzzy concepts, fuzzy classification is more suitable for the needs of biological classification. It has been applied to image recognition and other fields.