Implications of Minimal Length Trees

Abstract

Fitch, W. M. and T. F. Smith (Department of Physiological Chemistry, University of Wisconsin, Madison, Wisconsin 53706 and Department of Physics, Northern Michigan University, Marquette, Michigan 49855) 1982. Implications of minimal length trees. Syst. Zool., 31:68-75.-Trees with branch length assignments may be constrained so that the sum of branches connecting any two labelled tips of the tree is at least as large as a previously observed distance value. When the sum of all branch lengths is minimized under this constraint, one has a minimal length tree. Two conditions regarding minimal length trees, one proved necessary and the other conjectured to be sufficient, are presented. They provide. simple tests of minimality and raise problems in the interpretation of the individual branch values. [Minimal trees; Hamiltonian circuits; evolutionary rates; maximum parsimony.] One of the commonly employed criteria in systematics has been the minimization of the length of trees constructed from distance or dissimilarity data. This criterion is but one of a more general class in which hierarchical clusterings are evaluated by some objective function on the dissimilarity data. These objective functions have included statistical measures such as Chi-squared or the sum of deviations between the tree implied distances and the measured distances as well as various correlation coefficients between the tree-implied relations and the measured values. The popularity of minimizing the sum of branch lengths as an objective function arises from two considerations. The first is its conceptual simplicity, the fact that one is minimizing over an understood model parameter, such as the number of genetic fixation events. The second is that there have been reasonable, if not compelling, epistemological arguments put forward (Farris, 1978; Fitch, 1971) including the recognition that the amount of historical change cannot be less than the amount required to account for observed differences. Unfortunately, there are no proven, mathematically efficient methods for generating both the tree topology and the minimal sum of its branch lengths given no other data than the dissimilarity measures. Note the existence of two independent but interrelated problems. One is to determine the minimal tree length given a specific tree topology. Two is to determine which tree topology gives the lowest minimum. The best that can currently be done rigorously is to assign the minimum branch lengths for a tree of given branching topology using Linear Programming (Waterman et al., 1977). Yet even here there is the problem of practical computer time limitations which generally restrict the Linear Programming to trees of less than 30 terminal taxa. There are, of course, various heuristic algorithms for trying to find the tree of minimal length directly from a set of inter-taxa distances, the most commonly employed being the Wagner distance algorithm (Farris, 1972). In this study, we examine the properties of a tree that is of minimal length given the branching topology (problem one above). Two conditions for minimal length (one of which is also conjectured to be sufficient) are proven necessary and discussed. These properties thus may be used to test a claim of the minimality of any tree. Because of the widespread use and debate surrounding the minimal length criterion, we have carefully stated both our definitions and theorems. This has been done with the minimum of mathematical jargon but with every attempt at maxi-