These notes are not necessarily an accurate representation of what happened in class. The notes written before class say what I think I should say. The notes written after class way what I wish I said. 9.2 Overview The goal of this lecture is to present an approach of Stone and Griffing [SG09] to reconstructing evolutionary trees. They take the distances between all pairs of leaves in a tree and use this to reconstruct the tree. There are many other algorithms for doing this. I will add some references when I revise the notes. I present this approach both because it reveals the power of the second eigenvector of the Laplacian, and because I think it can be made to get around some of the deficits of some of the other algorithms. 9.3 Idealized Evolutionary Trees I will now give an idealized view of how we could reconstruct the tree of life from the DNA of existing organisms. Evolutionary theory tells us that species split off from each other through mutation. Thus, we should be able to arrange the species that have existed into a very large tree with one vertex for each species. The root should be the first organism, and the children of eac
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