Some mathematical properties of branching trees with application to the respiratory system
A number of biological branching systems, such as the bronchial and pulmonary arterial trees, are being investigated in an ongoing study in order to define their physiological properties. The technique involves the description of branching trees by the use of hierarchical systems of ordering, especially those described by Horsfield and by Strahler. During this work some mathematical properties of branching trees were demonstrated and these are described in this paper.
Unable to display preview. Download preview PDF.
- Gravelius, H. 1914.Flusskunde, band 1. Berlin and Leipzig: F. G. Goschenesche Verlagshandlung.Google Scholar
- Horsfield, K. and G. Cumming. 1967. “Angles of branching and diameters of branches in the human bronchial tree.”Bull. Math. Biophys.,29, 245–259.Google Scholar
- — and —. 1968. “Morphology of the bronchial tree in man.”J. Appl. Physiol.,24, 373–383.Google Scholar
- —, G. Dart, D. E. Olson, G. Filley and G. Cumming. 1971. “Models of the human bronchial tree.”J. Appl. Physiol.,31, 207–217.Google Scholar
- Horton, R. E. 1945. “Erosional development of streams and their drainage basins; hydrophysical approach to quantitative morphology.”Bull. Geogrl Soc. Am.,56, 275–370.Google Scholar
- Leopold, L. B. and J. P. Miller, 1956. “Ephemeral streams—hydraulic factors and their relation to the drainage net.”U.S. Geological Survey Professional Paper 282-A.Google Scholar
- Scheidegger, A. E. 1965. “The algebra of stream-order numbers.”U.S. Geological Survey Professional Paper 525-B, pp. B187–189.Google Scholar
- Singhal, S., R. Henderson, K. Horsfield, K. Harding and G. Cumming. 1973. “Morphometry of the human pulmonary arterial tree.”Circ. Res.,33, 190–197.Google Scholar
- Strahler, A. N. 1953. “Revision of Horton's quantitative factors in erosional terrain.”Trans. Am. Geophysical Union,34, 345.Google Scholar
- Strahler, A. N. 1957. “Quantitative analysis of watershed geomorphology.”Trans. Am. Geophysical Union,38, 913–920.Google Scholar
- Weibel, E. R. 1963.Morphometry of the human lung. Berlin: Springer-Verlag.Google Scholar
- Woldenberg, M. J. 1967. “Geography and properties of surfaces.”Harvard Papers in Theoretical Geography,1, 95–189.Google Scholar