Combinatorial analysis of ramified patterns and computer imagery of trees

Abstract

Herein is presented a new procedural method for generating images of trees. Many other algorithms have already been proposed in the last few years focusing on particle systems, fractals, graftals and L-systems or realistic botanical models. Usually the final visual aspect of the tree depends on the development process leading to this form. Our approach differs from all the previous ones. We begin by defining a certain "measure" of the form of a tree or a branching pattern. This is done by introducing the new concept of ramification matrix of a tree. Then we give an algorithm for generating a random tree having as ramification matrix a given arbitrary stochastic triangular matrix. The geometry of the tree is defined from the combinatorial parameters implied in the analysis of the forms of trees. We obtain a method with powerful control of the final form, simple enough to produce quick designs of trees without loosing in the variety and rendering of the images. We also introduce a new rapid drawing of the leaves. The underlying combinatorics constitute a refinment of some work introduced in hydrogeology in the morphological study of river networks. The concept of ramification matrix has been used very recently in physics in the study of fractal ramified patterns.