Comparative performances of new and existing indices of crown asymmetry: an evaluation using tall trees of Eucalyptus pilularis (Smith)

Abstract

Over the past 50 years, crown asymmetry of forest trees has been evaluated through several indices constructed from the perspective of projected crown shape or displacement but often on an ad hoc basis to address specific objectives related to tree growth and competition, stand dynamics, stem form, crown structure and treefall risks. Although sharing some similarities, these indices are largely incoherent and non-comparable as they differ not only in the scale but also in the direction of their values in indicating the degree of crown asymmetry. As the first attempt at devising normative measures of crown asymmetry, we adopted a relative scale between 0 for perfect symmetry and 1 for extreme asymmetry. Five existing crown asymmetry indices (CAIs) were brought onto this relative scale after necessary modifications. Eight new CAIs were adapted from measures of circularity for digital images in computer graphics, indices of income inequality in economics, and a bilateral symmetry indicator in plant leaf morphology. The performances of the 13 CAIs were compared over different numbers of measured crown radii for 30 projected crowns of mature Eucalyptus pilularis trees through benchmarking statistics and rank order correlation analysis. For each CAI, the index value based on the full measurement of 36 evenly spaced radii of a projected crown was taken as the true value in the benchmarking process. The index (CAI ₁₃) adapted from the simple bilateral symmetry measure proved to be the least biased and most precise. Its performance was closely followed by that of three other CAIs. The minimum number of crown radii that is needed to provide at least an indicative measure of crown asymmetry is four. For more accurate and consistent measures, at least 6 or 8 crown radii are needed. The range of variability in crown morphology of the trees under investigation also needs to be taken into consideration. Although the CAIs are from projected crown radii, they can be readily extended to individual tree crown metrics that are now commonly extracted from LiDAR and other remotely sensed data. Adding a normative measure of crown asymmetry to individual tree crown metrics will facilitate the process of big data analytics and artificial intelligence in forestry wherever crown morphology is among the factors to be considered for decision making in forest management.

Keywords

Projected crown shape / Circularity / Inequality / Bilateral symmetry / Rank order / Eucalyptus pilularis

References

[1] Aakala T, Shimatani K, Abe T, Kubota Y, Kuuluvainen T. Crown asymmetry in high latitude forests: disentangling the directional effects of tree competition and solar radiation. Oikos, 2016, 125(7): 1035-1043.

[2] Allison PD. Measures of inequality. Am Sociol Rev, 1978, 43: 865-880.

[3] Atkinson AB. On the measurement of inequality. J Econ Theory, 1970, 2: 244-263.

[4] Bar-Ness YD, Kirkpatrick JB, McQuillan PB. Crown structure differences and dynamics in 100-year-old and old-growth Eucalyptus obliqua trees. Aust For, 2012, 75(2): 120-129.

[5] Bendel RB, Higgins SS, Teberg JE, Pyke DA. Comparison of skewness coefficient, coefficient of variation, and Gini coefficient as inequality measures within populations. Oecologia, 1989, 78(3): 394-400.

[6] Bi H (1989) Growth of Pinus radiata (D. Don) stands in relation to intra- and inter-specific competition. PhD Thesis, The University of Melbourne, Melbourne, Australia

[7] Binkley D, Kashian DM, Boyden S, Kaye MW, Bradford JB, Arthur MA, Fornwalt PJ, Ryan MG. Patterns of growth dominance in forests of the Rocky Mountains, USA. For Ecol Manag, 2006, 236(2–3): 193-201.

[8] Bivand R, Rundel C, Pebesma E, Stuetz R, Hufthammer KO, Bivand MR (2017) Package ‘rgeos’, The Comprehensive R Archive Network (CRAN)

[9] Boland DJ, Brooker MIH, Chippendale GM, Hall N, Hyland BPM, Johnston RD, Kleinig DA, McDonald MW, Turner JD. Forest trees of Australia, 2006, Clayton: CSIRO Publishing

[10] Bowles S, Carlin W. Inequality as experienced difference: a reformulation of the Gini coefficient. Econ Lett, 2020, 186: 108789.

[11] Bredenkamp BV (1984) The CCT concept in spacing research-a review. In: Grey DC, Schönau APG, Schutz CJ (eds) Proceedings of the IUFRO symposium on site and productivity of fast-growing plantations, vol 30, pp 313–332

[12] Breunig R, Hutchinson DLA. Toggins WN. Small sample bias corrections for inequality indices. New econometric modeling research, 2008, New York: Nova Science Publishers.

[13] Brisson J. Neighborhood competition and crown asymmetry in Acer saccharum. Can J For Res, 2001, 31(12): 2151-2159.

[14] Brown PL, Doley D, Keenan RJ. Estimating tree crown dimensions using digital analysis of vertical photographs. Agric For Meteorol, 2000, 100: 199-212.

[15] Brüchert F, Gardiner B. The effect of wind exposure on the tree aerial architecture and biomechanics of Sitka spruce (Picea sitchensis, Pinaceae). Am J Bot, 2006, 93(10): 1512-1521.

[16] Cassidy M, Palmer G, Glencross K, Nichols JD, Smith RGB. Stocking and intensity of thinning affect log size and value in Eucalyptus pilularis. For Ecol Manag, 2012, 264: 220-227.

[17] Ceriani L, Verme P. The origins of the Gini index: extracts from Variabilità e Mutabilità (1912) by Corrado Gini. J Econ Inequal, 2012, 10(3): 421-443.

[18] Chattopadhyay B, De SK. Estimation of Gini index within pre-specified error bound. Econometrics, 2016 4 3 30

[19] Curtin RA. Dynamics of tree and crown structure in Eucalyptus obliqua. For Sci, 1970, 46(3): 321-328.

[20] Deltas G. The small-sample bias of the Gini coefficient: results and implications for empirical research. Rev Econ Stat, 2003, 85(1): 226-234.

[21] Di Ruberto C, Dempster A. Circularity measures based on mathematical morphology. Electron Lett, 2000, 36(20): 1691-1693.

[22] Dunham RA, Cameron AD. Crown, stem and wood properties of wind-damaged and undamaged Sitka spruce. For Ecol Manag, 2000, 135(1–3): 73-81.

[23] Engel M, Körner M, Berger U. Plastic tree crowns contribute to small-scale heterogeneity in virgin beech forests-An individual-based modeling approach. Ecol Model, 2018, 376: 28-39.

[24] Ferrante MR, Pacei S. Small sample bias corrections for entropy. Biostat Biom Open Access J, 2019 9 3 555765

[25] Fleck S, Mölder I, Jacob M, Gebauer T, Jungkunst HF, Leuschner C. Comparison of conventional eight-point crown projections with LIDAR-based virtual crown projections in a temperate old-growth forest. Ann For Sci, 2011, 68(7): 1173-1185.

[26] Florence RG. Variation in Blackbutt. Aust For, 1969, 33: 83-93.

[27] Florence RG. Ecology and silviculture of eucalypt forests, 1996, Collingwood: CSIRO 413

[28] Fox JC, Bi H, Ades PK. Spatial dependence and individual-tree growth models: I. Characterising spatial dependence. For Ecol Manag, 2007, 245(1–3): 10-19.

[29] Franco M. The influence of neighbours on the growth of modular organisms with an example from trees. Phil Trans R Soc B Biol Sci, 1986, 313: 209-225.

[30] Frosini BV. Approximation and decomposition of Gini, Pietra–Ricci and Theil inequality measures. Empir Econ, 2012, 43(1): 175-197.

[31] Getzin S, Wiegand K. Asymmetric tree growth at the stand level: random crown patterns and the response to slope. For Ecol Manag, 2007, 242(2–3): 165-174.

[32] Giles DE (2005) The bias of inequality measures in very small samples: some analytic results (No. 0514). Department of Economics, University of Victoria, Canada

[33] Grams TE, Andersen CP (2007) Competition for resources in trees: physiological versus morphological plasticity. In Progress in botany. Springer, Berlin, pp 356–381

[34] Greselin F, Pasquazzi L. Asymptotic confidence intervals for a new inequality measure. Commun Stat Simul Comput, 2009, 38(8): 1742-1756.

[35] Gupta AK, Nadarajah S. Handbook of beta distribution and its applications, 2004, New York: CRC Press

[36] Hajek P, Seidel D, Leuschner C. Mechanical abrasion, and not competition for light, is the dominant canopy interaction in a temperate mixed forest. For Ecol Manag, 2015, 348: 108-116.

[37] Han Q, Kabeya D, Saito S, Araki MG, Kawasaki T, Migita C, Chiba Y. Thinning alters crown dynamics and biomass increment within aboveground tissues in young stands of Chamaecyparis obtusa. J For Res, 2014, 19(1): 184-193.

[38] Haralick RM. A measure for circularity of digital figures. IEEE Trans Syst Man Cybern, 1974, 4: 394-396.

[39] Hastings JH, Ollinger SV, Ouimette AP, Sanders-DeMott R, Palace MW, Ducey MJ, Sullivan FB, Basler D, Orwig DA. Tree species traits determine the success of LiDAR-based crown mapping in a mixed temperate forest. Remote Sens, 2020 12 2 309

[40] Henson M, Smith HJ. Achievements in forest tree genetic improvement in Australia and New Zealand 1: Eucalyptus pilularis Smith tree improvement in Australia. Aust For, 2007, 70(1): 4-10.

[41] Herrera-Navarro AM, Jiménez Hernández H, Peregrina-Barreto H, Manríquez-Guerrero F, Terol-Villalobos IR. A new measure of circularity based on distribution of the radius. Computacióny Sistemas, 2013, 17(4): 515-526.

[42] Heshmati A (2004) Inequalities and their measurement. IZA discussion paper no. 1219. https://ssrn.com/abstract=571662

[43] Hess C, Härdtle W, Kunz M, Fichtner A, von Oheimb G. A high-resolution approach for the spatiotemporal analysis of forest canopy space using terrestrial laser scanning data. Ecol Evol, 2018, 8(13): 6800-6811.

[44] Hoover EM. The measurement of industrial localization. Rev Econ Stat, 1936, 18: 162-171.

[45] Hoover EM. Interstate redistribution of population, 1850–1940. J Econ Hist, 1941, 1(2): 199-205.

[46] Iiames JS, Cooter E, Schwede D, Williams J. A comparison of simulated and field-derived leaf area index (LAI) and canopy height values from four forest complexes in the southeastern USA. Forests, 2018 9 1 26

[47] Jasso G. On Gini’s mean difference and Gini’s index of concentration. Am Sociol Rev, 1979, 44(5): 867-870.

[48] Jucker T, Bouriaud O, Coomes DA. Crown plasticity enables trees to optimize canopy packing in mixed–species forests. Funct Ecol, 2015, 29(8): 1078-1086.

[49] Kinny M, McElhinny C, Smith G. The effect of gap size on growth and species composition of 15-year-old regrowth in mixed blackbutt forests. Aust For, 2012, 75(1): 3-15.

[50] Kio PRO. Relationships between asymmetry of the crown and the radial distribution of buttress flanges in some tropical timber species. Commonw For Rev, 1970, 49: 261-266.

[51] Kira T, Ogawa H, Sakazaki N. Intraspecific competition among higher plants I. Competition-yield-density interrelationship in regularly dispersed population. J Inst Poly Osaka City Univ, 1953, D4: 1-16.

[52] Krajicek JE, Brinkman KA, Gingrich SF. Crown competition-a measure of density. For Sci, 1961, 7(1): 35-42.

[53] Krůček M, Trochta J, Cibulka M, Král K. Beyond the cones: how crown shape plasticity alters aboveground competition for space and light—evidence from terrestrial laser scanning. Agric For Meteorol, 2019, 264: 188-199.

[54] Langel M, Tillé Y. Variance estimation of the Gini index: revisiting a result several times published. J R Stat Soc Ser A (Stat Soc), 2013, 176(2): 521-540.

[55] Lei B, Zhang G, Liu S, Liu X, Xi R, Wang X. Difference and cause analysis of crown shape of three tree species in different site conditions of Jinsha River Region. For Invent Plan, 2012, 37(2): 28-32. (in Chinese with English title and abstract)

[56] Lexerød NL, Eid T. An evaluation of different diameter diversity indices based on criteria related to forest management planning. For Ecol Manag, 2006, 222(1–3): 17-28.

[57] Long L, Nucci A. The Hoover index of population concentration: a correction and update. Prof Geogr, 1997, 49(4): 431-440.

[58] Longuetaud F, Piboule A, Wernsdörfer H, Collet C. Crown plasticity reduces inter-tree competition in a mixed broadleaved forest. Eur J For Res, 2013, 132(4): 621-634.

[59] Lorenz MO. Methods of measuring the concentration of wealth. Publ Am Stat Assoc, 1905, 9(70): 209-219.

[60] McPherson EG, Rowntree RA. Geometric solids for simulation of tree crowns. Landsc Urban Plan, 1988, 15(1–2): 79-83.

[61] Mead R. A relationship between individual plant-spacing and yield. Ann Bot, 1966, 30(2): 301-309.

[62] Meng SX, Rudnicki M, Lieffers VJ, Reid DE, Silins U. Preventing crown collisions increases the crown cover and leaf area of maturing lodgepole pine. J Ecol, 2006, 94(3): 681-686.

[63] Metz J, Seidel D, Schall P, Scheffer D, Schulze ED, Ammer C. Crown modeling by terrestrial laser scanning as an approach to assess the effect of aboveground intra-and interspecific competition on tree growth. For Ecol Manag, 2013, 310: 275-288.

[64] Montero RS, Bribiesca E. State of the art of compactness and circularity measures. Int Math Forum, 2009, 4(27): 1305-1335.

[65] Muneri A, Smith RGB, Armstrong M, Andrews M, Joe B, Dingle J, Dickson R, Nester M, Palmer G. The impact of spacing and thinning on growth, sawing characteristics and wood properties of 36-year-old Eucalyptus pilularis, 2003, Coffs Harbour: Internal Report, Forests NSW 72

[66] O’Connor AJ (1935) Forest research with specific reference to planting distances and thinning. In: British empire economic conference, p 30

[67] Olivier MD, Robert S, Fournier RA. Response of sugar maple (Acer saccharum, Marsh.) tree crown structure to competition in pure versus mixed stands. For Ecol Manag, 2016, 374: 20-32.

[68] Paletto A, Tosi V. Forest canopy cover and canopy closure: comparison of assessment techniques. Eur J For Res, 2009, 128(3): 265-272.

[69] Pont D (2016) Assessment of individual trees using aerial laser scanning in New Zealand radiata pine forests. PhD thesis, School of Forestry, University of Canterbury

[70] Pretzsch H. Canopy space filling and tree crown morphology in mixed-species stands compared with monocultures. For Ecol Manag, 2014, 327: 251-264.

[71] Purves DW, Lichstein JW, Pacala SW. Crown plasticity and competition for canopy space: a new spatially implicit model parameterized for 250 North American tree species. PLoS ONE, 2007 2 9 e870

[72] Rogerson PA. The Gini coefficient of inequality: a new interpretation. Lett Spat Resour Sci, 2013, 6(3): 109-120.

[73] Rouvinen S, Kuuluvainen T. Structure and asymmetry of tree crowns in relation to local competition in a natural mature Scots pine forest. Can J For Res, 1997, 25: 1876-1880.

[74] Seidel D, Leuschner C, Müller A, Krause B. Crown plasticity in mixed forests—quantifying asymmetry as a measure of competition using terrestrial laser scanning. For Ecol Manag, 2011, 261(11): 2123-2132.

[75] Sen AK. On economic inequality, enlarged edition with a substantial annexe by Foster JE and Sen AK, 1997, Oxford: Oxford University Press.

[76] Sharma M, Burkhart HE, Amateis RL. Spacing rectangularity effect on the growth of loblolly pine plantations. Can J For Res, 2002, 32(8): 1451-1459.

[77] Shi P, Zheng X, Ratkowsky D, Li Y, Wang P, Cheng L. A simple method for measuring the bilateral symmetry of leaves. Symmetry, 2018 10 4 118

[78] Shinozaki K, Kira T. The CD rule, its theory and practical uses. J Biol Osaka City Univ, 1961, 12: 69-82.

[79] Siemon GR, Wood GB, Forrest WG. Effects of thinning on crown structure in radiata pine. NZ J For Sci, 1976, 6(1): 57-66.

[80] Sillett SC, Goslin MN. Distribution of epiphytic macrolichens in relation to remnant trees in a multiple-age Douglas-fir forest. Can J For Res, 1999, 29(8): 1204-1215.

[81] Sokal RR, Rohlf FJ. Biometry: the principles and practice of statistics in biological research, 1981, New York: WH Freeman and Company 859

[82] Stanton R (1992) Eucalyptus plantations in New South Wales. Research Paper No. 15, Forestry Commission of New South Wales, Sydney, p 29

[83] Stojmenovic M, Jevremovic A, Nayak A (2013) Fast iris detection via shape based circularity. In: 2013 IEEE 8th conference on industrial electronics and applications (ICIEA), pp 747–752

[84] Stumpf KA (1993) The estimation of forest vegetation cover descriptions using a vertical densitometer. In: Joint inventory and biometrics working groups session at the SAF National Convention, Indianapolis, IN

[85] Teste FP, Lieffers VJ. Snow damage in lodgepole pine stands brought into thinning and fertilization regimes. For Ecol Manag, 2011, 261(11): 2096-2104.

[86] Theil H. Economics and information theory, 1967, Chicago: Rand McNally.

[87] Trochta J, Krůček M, Vrška T, Král K. 3D forest: an application for descriptions of three-dimensional forest structures using terrestrial LiDAR. PLoS ONE, 2017 12 5 e0176871

[88] Umeki K. Modeling the relationship between the asymmetry in crown display and local environment. Ecol Model, 1995, 82(1): 11-20.

[89] Umeki K. A comparison of crown asymmetry between Picea abies and Betula maximowicziana. Can J For Res, 1995, 25: 1876-1880.

[90] Umeki K. Effect of crown asymmetry on size-structure dynamics of plant populations. Ann Bot, 1997, 79(6): 631-641.

[91] Uria-Diez J, Pommerening A. Crown plasticity in Scots pine (Pinus sylvestris L.) as a strategy of adaptation to competition and environmental factors. Ecol Model, 2017, 356: 117-126.

[92] Vincent G, Harja D. Exploring ecological significance of tree crown plasticity through three-dimensional modelling. Ann Bot, 2008, 101(8): 1221-1231.

[93] Vovides AG, Berger U, Grueters U, Guevara R, Pommerening A, Lara-Domínguez AL, López-Portillo J. Change in drivers of mangrove crown displacement along a salinity stress gradient. Funct Ecol, 2018, 32(12): 2753-2765.

[94] Wade JE, Hewson EW. Trees as a local climatic wind indicator. J Appl Meteorol, 1979, 18(9): 1182-1187.

[95] Wei T, Simko V, Levy M, Xie Y, Jin Y, Zemla J. Package ‘corrplot’. Statistician, 2017, 56: 316-324.

[96] Weiner J. Size hierarchies in experimental populations of annual plants. Ecology, 1985, 66(3): 743-752.

[97] Weiner J, Solbrig OT. The meaning and measurement of size hierarchies in plant populations. Oecologia, 1984, 61(3): 334-336.

[98] West PW. Calculation of a growth dominance statistic for forest stands. For Sci, 2014, 60(6): 1021-1023.

[99] West PW, Smith RGB. Inter-tree competitive processes during early growth of an experimental plantation of Eucalyptus pilularis in sub-tropical Australia. For Ecol Manag, 2019, 451: 117450.

[100] White MJ. Segregation and diversity measures in population distribution. Population Index, 1986, 52(2): 198-221.

[101] Wieland T. REAT: a regional economic analysis toolbox for R. Region, 2019, 6(3): R1-R57.

[102] Wooldridge GL, Musselman RC, Sommerfeld RA, Fox DG, Connell BH. Mean wind patterns and snow depths in an alpine-subalpine ecosystem as measured by damage to coniferous trees. J Appl Ecol, 1996, 33(1): 100-108.

[103] Xiao B, Wang G. Shape circularity measure method based on radial moments. J Electron Imaging, 2013 22 3 033022

[104] Xu K (2004) How has the literature on Gini’s index evolved in the past 80 years? Dalhousie University, Economics working paper

[105] Xu W, Su Z, Feng Z, Xu H, Jiao Y, Yan F. Comparison of conventional measurement and LiDAR-based measurement for crown structures. Comput Electron Agric, 2013, 98: 242-251.

[106] Young TP, Hubbell SP. Crown asymmetry, treefalls, and repeat disturbance of broad-leaved forest gaps. Ecology, 1991, 72(4): 1464-1471.

[107] Young TP, Perkocha V. Treefalls, crown asymmetry, and buttresses. J Ecol, 1994, 82(2): 319-324.

[108] Zhang YH, Li Y, Bi H. Converting diameter measurements of Pinus radiata taken at different breast heights. Aust For, 2015, 78(1): 1-5.