The Combinatorics of Reconstructing Evolutionary Trees L. A. Sz´ ekely,* P. L. Erd˝os, M. A. Steel ABSTRACT: We survey here results and problems from the reconstruction theory of evolutionary trees, which involve enumeration and inversion. 1. Introduction Since the work of Darwin, there has been a dream of biologists: to reconstruct the tree of evolution of living things. That tree could be the only scientific basis for classification. In the last two decades the dramatic progress in molecular biology (reading long segments of genetic sequences) led to a new field, the theory of molecular evolution. One assumes that the process of evolution is described by a tree, in which no degree exceeds 3, since evolutionary events are too rare to coincide. In this tree the leaves denote existing species represented by corresponding segments of aligned DNA sequences, the unlabelled branching vertices may denote unknown extinct ancestors; since fossils do not keep records of the DNA sequence. For a given set of existing species, we define their true tree by taking the subtree induced by them in the tree describing the process of evolution and undoing the vertices of degree two. We term any binary tree, in which leaves are labelled by the species and the branching vertices are unlabelled, an evolutionary tree. The very problem of reconstruction may be put in this way: given a set of species with corresponding segments of aligned DNA sequences, select the true tree from the set of possible evolutionary trees. In this paper we assume that every bit of the aligned DNA sequence is one of the four nucleotides, A (Adenine), G (Guanine), C (Cytosine), T (Thymine); i.e. we neglect insertions and deletions. Biologists also would like to add a root r to denote a common ancestor and the direction of the evolution. This root r may subdivide an edge of the true tree or may be attached to a vertex of the true tree. However, if you have a procedure to * Research of the first author was supported by the A. v. Humboldt-Stiftung while he was at the Rheinische Friedrich-Wilhelms Universit¨ at, Bonn, and the U. S. Office of Naval Research under the contract N-0014-91-J-1385 1