Abstract
Statistical properties of topological binary trees are studied on the basis of the distribution of segments in relation to centrifugal order. Special attention is paid to the mean of this distribution in a tree as it will be used as a measure of tree topology. It will be shown how the expectation of the mean centrifugal order depends both on the size of the tree and on the mode of growth in the context of modelling the growth of tree structures. Observed trees can be characterized by their mean orders and procedures are described to find the growth mode that optimally corresponds to these data. The variance structure of the mean-order measure appears to be a crucial factor in these fitting procedures. Examples indicate that mean-order analysis is an accurate alternative to partition analysis that is based on the partitioning of segments over sub-tree pairs at branching points.
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Literature
Abramowitz, M., and I. A. Stegun (Eds). 1965.Handbook of Mathematical Functions. New York: Dover.
Berry, M. and D. Pymm. 1981. “Analysis of Neural Networks”. InAdvances in Physiological Science, Vol. 30.Neural Communications and Control, G. Szekely, E. Labos and S. Damjanovitch (Eds). 28th International Congress of Physiological Science, Budapest, 1980. Budapest: Akadémiai Kiadó. Oxford: Pergamon Press.
Bevington, P. R. 1969.Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill.
Dacey, M. F. and W. C. Krumbein. 1976. “Three Growth Models for Stream Channel Networks”.J. Geol. 84, 153–163.
Harding, E. F. 1971. “The Probabilities of Rooted Tree-Shapes Generated by Random Bifurcation”.J. appl. Prob. 3, 44–77.
Hollingworth, T. and M. Berry. 1975. “Network Analysis of Dendritic Fields of Pyramidal Cells in Neocortex and Purkinje Cells in the Cerebellum of the Rat”.R. Soc. London Phil. Trans. B270, 227–264.
Horsfield, K., M. J. Woldenberg and C. L. Bowes. 1987. “Sequential and Synchronous Growth Models Related to Vertex Analysis and Branching Ratio”.Bull. math. Biol. 49, 413–430.
Jarvis, R. S. 1972. “New Measure of the Topologic Structure of Dendritic Drainage Networks”.Water Resources Res. 8, 1265–1271.
— and A. Werrity. 1975. “Some Comments on Testing Random Topology Stream Network Models”.Water Resources Res. 11, 309–318.
Ley, K., A. R. Pries and P. Gaehtgens. 1986. “Topological Structure of Rat Mesenteric Microvessel Networks”.Microvascular Res. 32, 313–332.
Liao, K. H. and A. E. Scheidegger. 1968. “A Computer Model for Some Branching-Type Phenomena in Hydrology”.Int. Assoc. Sci. Hydrology Bull. 13, 5–13.
Press, W. H., B. P. Flannery, S. A. Teukolsky and W. T. Vettering (Eds.), 1986.Numerical Recipes, The Art of Scientific Computing. Cambridge University Press.
Sadler, M. and M. Berry. 1983. “Morphometric Study of the Development of Purkinje Cell Dendritic Trees in the Mouse Using Vertex Analysis”.J. Microsc. 131, 341–354.
Shreve, R. L. 1966. “Statistical Law of Stream Numbers”.J. Geol. 74, 17–37.
Stuermer, C. A. O. 1984. “Rules for Retinotectal Terminal Arborizations in the Goldfish Optic Tectum”.J. comp. Neurol. 229, 214–232.
Triller, A. and H. Korn. 1986. “Variability of Axonal Arborizations Hides Simple Rules of Construction: A Topological Study from HRP Intracellular Injections”.J. Comp. Neurol. 253, 500–513.
Uylings, H. B. M., P. McConnell, A. Ruiz-Marcos, J. Van Pelt and R. W. H. Verwer. 1983. “Differential Changes in Dendritic Patterns of Pyramidal and Non-Pyramidal Neurons After Undernutrition and its Subsequent Rehabilitation”. InThe Cell Biology of Plasticity, Fidia Research Series—Frontiers in Neuroscience. Abst. Bk 1, 295–297.
—, J. van Pelt and R. W. H. Verwer. 1989. “Topological Analysis of Individual Neurons”. InComputer Techniques in Neuroanatomy, J. J. Capowski, (Ed.), pp. 215–240. New York: Plenum Press.
Vannimenus, J., B. Nickel and V. Hakim. 1984. “Models of Cluster Growth on the Caley Tree”.Phys. Rev. B30, 391–399.
Van Pelt, J. and R. W. H. Verwer. 1983. “The Exact Probabilities of Branching Patterns under Terminal and Segmental Growth Hypotheses”.Bull. math. Biol. 45, 269–285.
— and — 1984 “New Classification Methods of Branching Patterns”.J. Microsc. 136, 23–34.
— and — 1985 “Growth Models (Including Terminal and Segmental Branching) for Topological Binary Trees”.Bull. math. Biol.,47, 323–336.
— and —. 1986. “Topological Properties of Binary Trees Grown With Order-Dependent Branching Probabilities”.Bull. math. Biol. 48, 197–211.
—. and H. B. M. Uylings. 1986. “Application of Growth Models to the Topology of Neuronal Branching Patterns”.J. neurosci. Meth. 18, 153–165.
—, and —. 1987. “Mean Centrifugal Order as a Measure for Branching Pattern Topology”.Acta Stereol.,6, 393–397.
Verwer, R. W. H. and J. Van, Pelt. 1983. “A New Method For the Topological Analysis of Neuronal Tree Structures”.J. neurosci. Meth. 8, 335–351.
— and —. 1986. “Descriptive and Comparative Analysis of Geometrical Preperties of Neuronal Tree Structures”.J. neurosci. Meth. 18 179–206.
— and A. J. Noest. 1987. “Parameter Estimation in Topological Analysis of Binary Tree Structures”.Bull. math. Biol. 49, 363–378.
Werner, C. and J. S. Smart. 1973 “Some New Methods of Topologic Classification of Channel Networks”.Geog. Analysis 5, 271–295.
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van Pelt, J., Verwer, R.W.H. & Uylings, H.B.M. Centrifugal-order distributions in binary topological trees. Bltn Mathcal Biology 51, 511–536 (1989). https://doi.org/10.1007/BF02460088
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DOI: https://doi.org/10.1007/BF02460088