Fu Y X, Li W H
Center for Demographic and Population Genetics, University of Texas, Houston 77225.
Math Biosci. 1992 May;109(2):201-28. doi: 10.1016/0025-5564(92)90045-x.
An analytical method is presented for constructing linear invariants. All linear invariants of a k-species tree can be derived from those of (k-1)-species trees using this method. The new method is simpler than that of Cavender, which relies on numerical computations. Moreover, the new method provides a convenient tool to study the relationships between linear invariants of the same tree or of different trees. All linear invariants of trees of up to five species are derived in this study. For four species, there are 16 independent linear invariants for each of the three possible unrooted trees, 14 of which are shared by two unrooted trees and 12 of these are shared by all three unrooted trees; the last types of linear invariants can be used to construct tests on the assumptions about nucleotide substitutions. The number of linear invariants for a tree is found to increase rapidly with the number of species.
提出了一种构建线性不变量的分析方法。利用该方法，k物种树的所有线性不变量都可以从（k - 1）物种树的线性不变量推导得出。新方法比依赖数值计算的卡文德方法更简单。此外，新方法为研究同一棵树或不同树的线性不变量之间的关系提供了便利工具。本研究推导了多达五个物种的树的所有线性不变量。对于四个物种，三种可能的无根树中每种都有16个独立的线性不变量，其中14个为两种无根树所共有，12个为所有三种无根树所共有；最后这种类型的线性不变量可用于构建关于核苷酸替换假设的检验。发现树的线性不变量数量随着物种数量的增加而迅速增加。
