Properties of the nearest neighbor interchange metric for trees of small size

Abstract

The crossover or nearest neighbor interchange metric has been proposed for use in numerical taxonomy to obtain a quantitative measure of distance between classifications that are modeled as unrooted binary trees with labeled leaves. This metric seems difficult to compute and its properties are poorly understood. A variant called the closest partition distance measure has also been proposed, but no efficient algorithm for its computation has yet appeared and its relationship to the nearest neighbor interchange metric is incompletely understood. I investigate four conjectures concerning the nearest neighbor interchange and closest partition distance measures and establish their validity for trees with as many as seven labeled vertices. For trees in this size range the two distance measures are identical. If a certain decomposition property holds for the nearest neighbor interchange metric, then the two distance measures are also identical at small distances for trees of any size.