The Number of Evolutionary Steps on Random and Minimum Length Trees for Random Evolutionary Data

Abstract

A model of evolutationarily uninformative data is derived and two separate character state distributions, one with two states (0, 1) and one with missing-value data (0, 1, {0, 1}), are obtained. The expectation of number of steps on random trees is derived for both types of data and the variance in number of steps is derived for missing-value data. It is conjectured that the number of steps on random trees for these data should be asymptotically normal. Computer simulation is used to find approximations for the expected number and variance in number of steps of minimum length trees for both types of random evolutionary data. In both cases, minimum length trees are shorter than random trees and, although the lengths diverge in number, they converge in proportion. The length of minimum length trees is frequently used to evaluate the goodness of the tree as a representation of the data and the goodness of the data for constructing a plausible hypothesis of evolutionary relationships. In turn, as the length of a minimum length tree increases, confidence in the resulting tree decreases. The distributions of lengths of minimum length trees are of interest to determine if any information on phylogenetic branching is present in the data. The expected length of random trees can be used as a measure to compare to the lengths of minimum length trees for real data sets.