On the effect of measures for comparing trees on the representation of treespace

The multidimensional space determined by distances between trees (as measured with various methods) is often reduced and projected with multidimensional scaling to visually represent the differences between trees in the possible "treespace". This paper discusses the influence of 18 alternative measures of distance on mapping the treespace for all possible trees (or a large sample thereof) when trees of different degrees of resolution are included. Measures of distance appropriate for such mapping are expected to produce (hyper)spherical mappings, with resolved trees in the outer layer, less resolved trees in inner layers and the bush situated in the middle of the diagram. Measures of tree comparison that rescale the values by the observed (rather than potential) resolution produce an inversion of such an arrangement, with less resolved trees in the outer layers. Additionally, some measures are shown to be strongly influenced by tree shape, so that trees of certain shapes end up being situated at specific depths of the diagram (which may become so distorted as to not even look like a hypersphere).