Probabilistic Distributions for Acacia Mearnsii De Wild Total Height and the Influence of Environmental Factors


This paper discusses the hypothesis of height distribution on a forest stand of Acacia mearnsii De Wild, known as black wattle. It remains constant at varied growing environments and, in addition, they are not influenced by age factor. The Wakeby equation was applied. The research was carried out in a black wattle stand at varied age levels and over two different agroecological regions where plantations are found: Serra do Sudeste and Encosta do Sudeste, Rio Grande do Sul State, Brazil. It was observed that as the age rises there is an increase in the stand total height; while the number of trees decreases for the lower classes, it increases for the upper ones. This resulted in lengthening of the curve tail to the left and mode shift to the right, generating negative asymmetrical curves. Two types of height distribution were found: the sharp increase of probability in a specific class and some similar probabilities in successive classes. The distribution curves between the cultivation areas were statistically different and therefore the height distribution was dependent of environment.

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Sanquetta, C. , Behling, A. , Pelissari, A. , Corte, A. , Netto, S. and Simon, A. (2014) Probabilistic Distributions for Acacia Mearnsii De Wild Total Height and the Influence of Environmental Factors. Journal of Applied Mathematics and Physics, 2, 1-10. doi: 10.4236/jamp.2014.23001.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] K. Pearson, “On the Systematic Fitting of Curves to Observations and Measurements, Parts I and II,” Biometrika, Vol. 1, No. 3, 1902, pp. 265-303.
[2] R. A. Fisher, “On the Mathematical Foundations of Theoretical Statistics,” Philosophical Transactions of the Royal Society of London, Vol. 222, No. 594-604, 1922, pp. 309-368.
[3] S. D. Dubey, “Some Percentile Estimators for Weibull Parameters,” Technometrics, Vol. 9, No. 1, 1967, pp. 119-129.
[4] R. L. Bailey and T. R. Dell, “Quantifying Diameter Distributions with the Weibull Function,” Forest Science, Vol. 19, No. 2, 1973, pp. 97-104.
[5] G. W. Smalley and R. L. Bailey, “Yield Tables and Stand Structure for Loblolly Pine Plantations in Tennessee, Alabama, and Georgia Highlands,” Department of Agriculture, Forest Service, Southern Forest Experiment Station, New Orleans, 1974.
[6] J. L. Clutter and B. J. Allison, “A Growth and Yield Model for Pinus radiata in New Zealand for Tree and Stand Simulation,” Royal College of Forestry, Stockholm, 1974.
[7] J. L. Clutter, W. R. Harms, G. H. Brister and J. W. Rhenney, “Stand Structure and Yields of Site-Prepared Loblolly Pine Plantations in the Lower Coastal Plain of the Carolinas,” Gergia Forest Research Council, Atlanta, 1984.
[8] Q. V. Cao, “Predicting Parameters of a Weibull Function for Modeling Diameter Distribution,” Forest Science, Vol. 50, 2004, pp. 682-685.
[9] M. Palahi, T. Pukkala and A. Trasobares, “Modelling the Diameter Distribution of Pinus sylvestris, Pinus nigra and Pinus halepensis Forest Stands in Catalonia Using the Truncated Weibull Function,” Forestry, Vol. 79, No. 5, 2006, pp. 553-562.
[10] L. C. Jiang and J. R. Brooks, “Predicting Diameter Distributions for Young Longleaf Pine Plantations in Southwest Georgia,” Southern Journal of Applied Forestry, Vol. 33, 2009, pp. 25-28.
[11] S. Andrasev, M. Bobinac and S. Orlovic, “Diameter Structure Models of Black Poplar Selected Clones in the Section Aigeiros (Duby) Obtained by the Weibull Distribution,” Sumarski List, Vol. 133, No. 11-12, 2009, pp. 589-603.
[12] R. Petrás, J. Mecko and V. Nociar, “Diameter Structure of the Stands of Piplar Clones,” Journal of Forest Science, Vol. 56, No. 4, 2010, pp. 165-170.
[13] A. C. Carretero and E. T. Alvarez, “Modelling Diameter Distributions of L. Stands in ‘Los Alcornocales’ Natural Park (Cádiz-Málaga, Spain) by Using the Two Parameter Weibull Functions,” Forest Systems, Vol. 22, No. 1, 2013, pp. 15-24.
[14] S. R. Lindsay, G. R. Wood and R. C. Woollons, “Modelling the Diameter Distribution of Forest Stands Using the Burr Distribution,” Journal of Applied Statistics, Vol. 23, No. 6, 1996, pp. 609-619.
[15] J. H. Gove, M. J. Ducey, W. B. Leak and L. Zhang, “Rotated Sigmoid Structures in Managed Uneven-Aged Northern Hardwood Stands: A Look at the Burr type III Distribution,” Forestry, Vol. 81, No. 2, 2008, pp. 161-176.
[16] J. L. Clutter and F. A. Bennett, “Diameter Distributions in Old-Field Slash Pine Plantations,” Georgia Forest Research Council, 1965, pp. 1-9.
[17] F. A. Bennett and J. L. Clutter, “Multiple-Product Yield Estimates for Unthinned Slash Pine Plantations—Pulpwood, Saw Timber, Gum,” Gergia Forest Research Council, Atlanta, 1968.
[18] J. D. Lenhart and J. L. Clutter, “Cubic-Foot Yield Tables for Old-Field Loblolly Pine Plantations in the Georgia Piedmont,” Gergia Forest Research Council, Atlanta, 1971.
[19] H. E. Burkhart and M. R. Strub, “A Model for Simulation of Planted Loblolly Pine Stands,” In: J. Fries, Ed., Growth Models for Tree and Stand Simulation, Royal College of Forestry, Stockholm, 1974, pp. 128-135.
[20] F. A. Bennett, F. T. Lloyd, B. F. Swindel and E. W. Whitehorne, “Yields of Veneer and Associated Products from Unthinned, Old-Field Plantations of Slash Pine in the North Florida and South Georgia Flatwoods,” Gergia Forest Research Council, Atlanta, 1978.
[21] J. J. Gorgoso-Varela, A. Rojo-Alboreca, E. Afif-Khouri and M. Barrio-Anta, “Modeling Diameter Distributions of Birch (Betula alba L.) and Pedunculate Oak (Quercus robur L.) Stands in Northwest Spain with the Beta Distribution,” Investigación Agraria, Sistemas y Recursos Forestales, Vol. 17, 2008, pp. 271-281.
[22] T. C. Nelson, “Diameter Distribution and Growth of Loblolly Pine,” Forest Science, Vol. 10, No. 1, 1964, pp. 105-114.
[23] W. L. Hafley, W. D. Smith and M. A. Buford, “A New Yield Prediction Model for Unthinned Loblolly Pine Plantations,” North Carolina State University, Raleigh, 1982.
[24] K. Rennolls and M. Wang, “A New Parameterization of Johnson’s SB Distribution with Application to Fitting Forest Tree Diameter data,” Canadian Journal of Forest Research, Vol. 35, No. 3, 2005, pp. 575-579.
[25] T. F. Fonseca, C. P. Marques and B. R. Parresol, “Describing Maritime Pine Diameter Distributions with Johnson’s SB Distribution Using a New All-Parameter Recovery Approach,” Forest Science, Vol. 55, No. 4, 2009, pp. 367-373.
[26] C. I. Bliss and K. A. Reinker, “A Log-Normal Approach to Diameter Distribution in Even-Aged Stands,” Forest Science, Vol. 10, 1964, pp. 350-360.
[27] H. T. Schreuder and W. L. Hafley, “A Useful Bivariate Distribution for Describing Stand Structure Os Tree Heights and Diameters,” Biometrics, Vol. 33, No. 3, 1977, pp. 471-478.
[28] C. M. Chen and D. V. Rose, “Direct and Indirect Estimation of Height Distributions in Even-Aged Stands,” Minnesota Forestry Research Notes, No. 267, 1978.
[29] V. P. Tewari and K. V. Gadow, “Modelling the Relationship between Tree Diameters and Heights Using SBB Distribution,” Forest Ecology and Management, Vol. 119, No. 1-3, 1999, pp. 171-176.
[30] J. Siipilehto, “Height Distributions of Scots Pine Sapling Stands Affected by Retained Tree and Edge Stand Competition,” Silva Fennica, Vol. 40, No. 3, 2006, pp. 473-486.
[31] S. A. Machado, R. G. M. Nascimento, E. P. Miguel, S. J. Téo and A. L. D. Augustynczik, “Distribution of Total Height, Transvrese área and Individual Volume for Araucaria angustifolia (Bert.) O. Kuntze,” Cerne, Vol. 16, No. 1, 2010, pp. 12-21.
[32] M. Wang, A. Upadhyay and L. J. Zhang, “Trivariate Distribution Modeling of Tree Diameter, Height, and Volume,” Forest Science, Vol. 56, No. 3, 2010, pp. 290-300.
[33] K. Tsogt, T. Zandraabal and C. S. Lin, “Diameter and Height Distributions of Natural Even-Aged Pine Forests (Pinus sylvestris) in Western Khentey, Mongolia,” Journal of Forest Science, Vol. 28, No. 1, 2013, pp. 29-41.
[34] K. Pearson, “Contributions to the Mathematical Theory of Evolution, II: Skew Variation in Homogeneous Material,” Philosophical Transactions of the Royal Society of London, Vol. 186, 1985, pp. 343-414.
[35] K. R. Pearson, “Skew Variation, a Rejoinder,” Biometrika, Vol. 4, No. 1-2, 1905, pp. 169-212.
[36] N. L. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distributions,” Applied Probability and Statistics, New York, 1995.
[37] J. S. Park and J. W. Jeon, “Maximum Likelihood Estimation of Wakeby Distribution,” Technical Report, Kwangju, 2000.
[38] A. Tarsitano, “Fitting Wakeby Model Using Maximum Likelihood,” Statistica e Ambiente, Messina, 21-23 September 2005, pp. 253-256.
[39] J. C. Houghton, “Birth of a Parent: The Wakeby Distribution for Modeling Flood Flow,” Water Resources Research, Vol. 14, No. 6, 1978, pp. 1105-1109.
[40] G. A. Griffiths, “A Theoretically Based Wakeby Distribution for Annual Flood Series,” Hydrological Sciences, Vol. 34, No. 3, 1989, pp. 231-248.
[41] D. S. Wilks and M. McKay, “Extreme-Value Statistics for Snowpack Water Equivalent in the Northeastern United States Using the Cooperative Observer Network,” Journal of Applied Meteorology, Vo. 35, No. 5, 1996, pp. 706-713.<0706:EVSFSW>2.0.CO;2
[42] J. S. Park, H. S. Jung, R. S. Kim and J. H. Oh, “Modelling Summer Extreme Rainfall Over the Korean Peninsula Using Wakeby Distributions,” International Journal of Climatology, Vol. 21, No. 11, 2001, pp. 1371-1384.
[43] T. Oztekin, “Wakeby Distribution for Representing Anual Extreme and Partial Duration Rainfall Series,” Meteorological Applications, Vol. 14, No. 4, 2007, pp. 381-387.
[44] B. Su, X. W. Kundzewiez and T. Jiang, “Simulation of Extreme Precipitation over the Yangtze River Basin Using Wakeby Distribution,” Theoretical and Applied Climatology, Vol. 96, No. 3-4, 2009, pp. 209-219.
[45] T. Fischer, B. Su, Y. Luo and T. Scholten, “Probability Distribution of Precipitation Extremes for Weather Index-Based Insurance in the Zhujiang River Basin, South China,” Journal of Hydrometeorology, Vol. 13, No. 3, 2012, pp. 1023-1037.
[46] C. Oliver and B. Larson, “Forest Stand Dynamics: Update Edition,” John Wiley, New York, 1996
[47] B. V. Barnes, D. R. Zak, S. R. Denton and S. H. Spurr, “Forest Ecology,” 4th Edition, John Wiley & Sons, New York, 1998.
[48] M. A. Herbert, “Fertilisation of Trees at Planting,” In: L. MacLennam, Ed., Annual Research Report, Institute for Commercial Forestry Research, Pietermaritzburg, 1991, pp. 81-91.
[49] A. P. G. Schounau and W. J. K. Aldworth, “Site Evaluation in Back Watle with Special Reference to Soil Factors,” South African Forestry Journal, No. 156, 1991, pp. 1-6.
[50] M. F. G. Rachwal, G. R. Curcio and R. A. Dedecek, “Caracterizacao do Desenvolvimento e Producao de Madeira da Acacia mearnsii aos 3 e 5 Anos de Idade em Solos Derivados de Micaxistos no Município de Piratini, RS,” Embrapa Florestas, Colombo, 1997.

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