Exploring the set-theoretical structure of objects by additive trees

Arrangements of feature sets that have been proposed to represent qualitative and quantitative variation among objects are shown to generate identical sets of set-symmetric distances. The set-symmetric distances for these feature arrangements can be represented by path lengths in an additive linear tree. Imperfect versions of these feature arrangements are proposed, which also are indistinguishable by the set-symmetric distance model. The distances for the imperfect versions can be represented by path lengths in an additive imperfectly linear tree. When dissimilarities are defined by the more general contrast model and a constant may be added to proximity data, then for both the perfect and imperfect arrangements an additive tree analysis obtains a perfect fit with an imperfectly linear tree. However, in the case of the contrast model also the distinction between the perfect and imperfect arrangements disappears in that also for the perfect arrangements the resulting tree need no longer be linear.The author is grateful to Mathieu Koppen for detailed comments on an earlier version of this article.