The Characterization of Topology: A Comparison of Four topological Indices for Rooted Binary Trees

The quantification of the topological features of binary trees has been applied in several branches of biology, from botany to neurobiology to animal behaviour. The methods available for quantifying tree topology differ, both in how they are applied and how they relate to one another. In this paper, I study the behaviour of four commonly used topological indices in relation to Shreveʹs random model for binary trees (Shreve D, 1966) and a variety of simple growth rules. The goals of these exercises include the following: (i) Derivation of expected values for each of the topological indices over a range of tree sizes (magnitudes) of relevance to biological trees. (ii) Derivation of confidence limits for these expected values. (iii) Calculation of pairwise correlation for all the indices from the Monte Carlo simulations. And (iv) to explore the relationships between each of the indices and to develop an understanding about what aspects of branching each of the different indices reflects. From these analyses I suggest that care needs to be taken when comparing different topological indices because they are poorly correlated with one another and because they all show high dependence on the size of the examined tree. Independent of such considerations, the use of the total pathlength (Pe) is advocated, because it shows consistent and easily characterized behaviour in relation to the random model and relatively robust behaviour in relation to the growth simulations.