Development of a mathematical method for classifying and comparing tree architecture using parameters from a topological model of a trifurcating botanical tree

This paper describes a model for the topological mapping of trifurcating botanical trees. The model was based on a system of modular units that represented the interconnectivity of shoot meristems (terminal segments) and internodes (internal segments) within whole plant canopies, organized with increasing centrifugal ordering. The model was capable of describing the dynamics of plant growth as expressed by changes in topological parameters over time. Preliminary calculations for experimental trees indicated that the model represents growth in a biologically sound manner. Methods are described for the calculation of the architecture parameters size, size-complexity, structural complexity, and tree asymmetry index (TAI). Parameter calculations were based on the mathematical principles developed for the classification of bifurcating dendrite trees, and were designed to both extract structural information, and to enable statistical comparison between trees of different size. Parameters were mathematically adjusted for trifurcation, and appeared to be able to represent quantitatively the architectural properties of tree structures. In addition to the calculation of the TAI for trifurcating trees, new methods were developed to enable comparisons to be made of the architectural complexity of trifurcating trees of differing size. These were based on the principle of the pair-wise comparison of the mean centrifugal order number (MCON) with respect to segments against highest order number. We argue and illustrate that this principle can be more informative than that of pair-wise comparison of the MCON against tree degree (topological size). Further improvements to this method were made by examining branching points (vertices) rather than segments (links) to calculate the MCON.