Direct reconstruction of phylogenetic trees by maximum likelihood methods is computationally prohibitive for trees with many taxa; however, by computing all trees for subsets of taxa of size m, we can infer the entire tree. In particular, if m = 2, the traditional distance-based methods such as neighbor-joining [Saitou and Nei, 1987] and UPGMA [Sneath and Sokal, 1973] are applicable. Under distance-based methods, 2-leaf subtrees are completely determined by the total length between each pair of leaves. We extend this idea to m leaves by developing the notion of m-dissimilarity [Pachter and Speyer, 2004]. By building trees on subsets of size m of the taxa and rinding the total length, we can obtain an m-dissimilarity map. We will explain the generalized neighbor-joining (GNJ) algorithm [Levy et al., 2005] for obtaining a phylogenetic tree with edge lengths from an m-dissimilarity map.

This algorithm is consistent: given an m-dissimilarity map DT that comes from a tree T, GNJ returns the correct tree. However, in the case of data that is “noisy”, e.g., when the observed dissimilarity map does not lie in the space of trees, the accuracy of GNJ depends on the reliability of the subtree lengths. Numerical methods may run into trouble when models are of high degree (Section 1.3); exact methods for computing subtrees, therefore, could only serve to improve the accuracy of GNJ. One family of such methods consists of algorithms for finding critical points of the ML equations as discussed in Chapter 15 and in [Hoşten et al., 2005].