Mathematical and computer modelling reports: Sequential and synchronous growth models related to vertex analysis and branching ratios

Two models of binary tree growth are examined in terms of the Strahler order branching ratio (Rb) and the types of vertex produced during growth, and their inter-relationship. The sequential growth model is that described by Van Pelt and Verwer (1985, Bull. math. Biol.47, 323-336) in which random growth occurs according to attributed probabilities on terminal or internal segments, one branch at a time. This model generates values of Rb >= 3. The synchronous growth model is new and permits more than one segment to branch at a time, again randomly with attributed probabilities. This model generates values of Rb >= 2 and in particular, when only terminal branching is permitted, gives 2 =< Rb < 3. Such a model might explain the branching in the human bronchial tree, in which 2.5 =< Rb =< 2.8. Our synchronous model is an alternative to the centrifugal-order-dependent sequential model of Van Pelt and Verwer (1986, Bull. math. Biol.48, 197-211).