Using Limit Value Constraint Theory to Better Understand the Self-Thinning Rule of Forest
Abstract
1. Introduction
2. Materials and Methods
2.1. Overview of the Study Area
2.2. Design of the Experimental Plots
2.3. Computation of the Growth Rate for G, H, and D
2.4. Analyses of the Interrelationships among G, H, and D
3. Data Analysis and Results
3.1. Relationships between Basal Area and Average Height
3.2. The Trend in Stand G/H to the First Limit Value
3.3. Trend in to the Second Limit Value
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Zhang, J.; Oliver, W.W.; Powers, R.F. Reevaluating the self-thinning boundary line for ponderosa pine (Pinus ponderosa) forests. Can. J. For. Res. 2013, 43, 963–971. [Google Scholar] [CrossRef]
- De Prado, D.R.; Martín, R.S.; Bravo, F.; de Aza, C.H. Potential climatic influence on maximum stand carrying capacity for 15 Mediterranean coniferous and broadleaf species. For. Ecol. Manag. 2020, 460, 117824. [Google Scholar] [CrossRef]
- Jack, S.B.; Long, J.N. Linkages between silviculture and ecology: An analysis of density management diagrams. For. Ecol. Manag. 1996, 86, 205–220. [Google Scholar] [CrossRef]
- Lei, X.D.; Tang, S.Z.; Fu, L.Y. Quantitative Evaluation of Forest Site Quality—Theory, Method and Application; Chinese Forestry Press: Beijing, China, 2019. [Google Scholar]
- Zeide, B. How to measure stand density. Trees 2005, 19, 1–14. [Google Scholar] [CrossRef]
- Burkhart, H.E. Comparison of maximum size-density relationships based on alternate stand attributes for predicting tree numbers and stand growth. For. Ecol. Manag. 2013, 289, 404–408. [Google Scholar] [CrossRef]
- Reineke, L.H. Perfecting a stand-density index for even-aged forests. J. Agric. Res. 1933, 46, 627–638. [Google Scholar]
- Yang, Y.; Titus, S.J. Maximum size-density relationship for constraining individual tree mortality functions. For. Ecol. Manag. 2002, 168, 259–273. [Google Scholar] [CrossRef]
- Vospernik, S.; Sterba, H. Do competition-density rule and self-thinning rule agree? Ann. For. Sci. 2015, 72, 379–390. [Google Scholar] [CrossRef]
- Burkhart, H.E.; Tomé, M. Modelling Forest Trees and Stands; Springer: Dordrecht, The Netherlands, 2012. [Google Scholar]
- Yoda, K.; Kira, T.; Ogawa, H.; Hozumi, K. Self-thinning in overcrowded pure stand under cultivated and natural conditions. J. Biol. Osaka City Univ. 1963, 14, 107–129. [Google Scholar]
- White, J.; Harper, J.L. Correlated changes in plant size and number in plant populations. J. Ecol. 1970, 58, 467–485. [Google Scholar] [CrossRef]
- West, G.B.; Brown, J.H.; Enquist, B.J. A general model for the origin of allometric scaling laws in biology. Science 1997, 276, 122–126. [Google Scholar] [CrossRef]
- West, G.B.; Brown, J.H.; Enquist, B.J. A general model for the structure and allometry of plant vascular systems. Nature 1999, 400, 664–667. [Google Scholar] [CrossRef]
- Gorham, E. Shoot height, weight and standing crop in relation to density of monospecific plant stands. Nature 1979, 279, 148–150. [Google Scholar] [CrossRef]
- White, J. Demographic Factors in Populations of Plants; University of California Press: Berkeley, CA, USA, 1980. [Google Scholar]
- White, J. The allometric interpretation of self-thinning rule. J. Theor. Biol. 1981, 89, 475–500. [Google Scholar] [CrossRef]
- Cousens, R.; Hutchings, M.J. The relationship between density and mean frond weight in monospecific seaweed stands. Nature 1983, 301, 240–241. [Google Scholar] [CrossRef]
- Charru, M.; Seynave, I.; Morneau, F.; Rivoire, M.; Bontemps, J.D. Significant differences and curvilinearity in the self-thinning relationships of 11 temperate tree species assessed from forest inventory data. Ann. For. Sci. 2012, 69, 195–205. [Google Scholar]
- Weller, D.E. A reevaluation of the—3/2 power rule of plant self-thinning. Ecol. Monogr. 1987, 57, 23–43. [Google Scholar] [CrossRef]
- Weller, D.E. Self-thinning exponent correlated with allometric measures of plant geometry. Ecology 1987, 68, 813–821. [Google Scholar] [CrossRef]
- Fang, J.Y. Self-thinning rule in plant population. Rural Eco-Environ. 1992, 2, 7–12, (In Chinese with English Abstract). [Google Scholar]
- Li, F.R. A review on stand density-about the 3/2 Power Law. For. Res. 1995, 8, 25–32, (In Chinese with English Abstract). [Google Scholar]
- Osawa, A. Inverse relationship of crown fractal dimension to self-thinning exponent of tree population: A hypothesis. Can. J. For. Res. 1995, 25, 1608–1617. [Google Scholar] [CrossRef]
- Kikuzawa, K. Theoretical relationships between mean plant size, size distribution and self-thinning under one-sided competition. Ann. Bot. 1999, 83, 11–18. [Google Scholar] [CrossRef][Green Version]
- Wu, C.Z.; Hong, W. A study on the self-thinning law of Chinese fir plantation. Sci. Silvae Sin. 2000, 36, 97–101, (In Chinese with English Abstract). [Google Scholar]
- Del Río, M.; Montero, G.; Bravo, F. Analysis of diameter-density relationships and self-thinning in non-thinned even-aged Scots pine stands. For. Ecol. Manag. 2001, 142, 79–87. [Google Scholar] [CrossRef]
- Luis, J.F.S.; Fonseca, T.F. The allometric model in the stand density management of Pinus pinaster Ait. Ann. For. Sci. 2004, 61, 807–814. [Google Scholar] [CrossRef]
- Pretzsch, H.; Biber, P. A reevaluation of Reineke’s rule and stand density index. For. Sci. 2005, 51, 304–320. [Google Scholar]
- Vanclay, J.K.; Sands, P.J. Calibrating the self-thinning frontier. For. Ecol. Manag. 2009, 259, 81–85. [Google Scholar] [CrossRef]
- Rivoire, M.; Moguedee, G.L. A generalized self-thinning relationship for multi-species and mixed-size forests. Ann. For. Sci. 2012, 69, 207–219. [Google Scholar] [CrossRef]
- Mohler, C.L.; Marks, P.L.; Sprugel, D.G. Stand structure and allometry of tree during self-thinning of pure stand. J. Ecol. 1978, 68, 598–614. [Google Scholar]
- Ogawa, K. Time-trajectory of mean phytomass and density during a course of self-thinning in a sugi (Cryptomeria japonica D. Don) plantation. For. Ecol. Manag. 2005, 214, 104–110. [Google Scholar] [CrossRef]
- Aguirre, A.; del Rio, M.; Condes, S. Intra- and inter-specific variation of the maximum size-density relationship along an aridity gradient in Iberian pinewoods. For. Ecol. Manag. 2018, 411, 90–100. [Google Scholar] [CrossRef]
- Andrews, C.; Weiskittel, A.; D’Amato, A.W.; Simons-Legaard, E. Variation in the maximum stand density index and its linkage to climate in mixed species forests of the North American Acadian Region. For. Ecol. Manag. 2018, 417, 90–102. [Google Scholar] [CrossRef]
- Zeide, B. Tolerance and self-tolerance of trees. For. Ecol. Manag. 1985, 13, 149–166. [Google Scholar] [CrossRef]
- Yang, S.; Burkhart, H.E. Estimation of carrying capacity in loblolly pine (Pinus taeda L.). For. Ecol. Manag. 2017, 385, 167–176. [Google Scholar] [CrossRef]
- Zeide, B. Analysis of the—3/2 power rule of plant self-thinning. For. Sci. 1987, 33, 517–537. [Google Scholar]
- Sackville Hamilton, N.R.; Matthew, C.; Lemaire, G. In defence of the—3/2 boundary rule: A re-evaluation of self-thinning concepts and status. Ann. Bot. 1995, 76, 569–577. [Google Scholar] [CrossRef]
- Zhang, L.J.; Bi, H.Q.; Jeffrey, H.G.; Linda, S.H. A comparison of alternative methods for estimating the self-thinning boundary line. Can. J. For. Res. 2005, 35, 1507–1514. [Google Scholar] [CrossRef]
- Possato, E.L.; Calegario, N.; Nogueira, G.S.; Melo, E.D.; Alves, J.D. Estimate of stand density index for Eucalyptus urophylla using different fit methods. Rev. Arvore 2016, 40, 921–929. [Google Scholar] [CrossRef]
- Trouvé, R.; Nitschke, C.R.; Robinson, A.P.; Baker, P.J. Estimating the self-thinning line from mortality data. For. Ecol. Manag. 2017, 402, 122–134. [Google Scholar] [CrossRef]
- Meng, J.H. A comparison of different methods for fitting the self-thinning equation. J. Beijing For. Univ. 2019, 41, 58–68, (In Chinese with English Abstract). [Google Scholar]
- Westoby, M. The self-thinning rule. Adv. Ecol. Res. 1984, 14, 167–225. [Google Scholar]
- Wilson, F.G. Thinning as an orderly discipline: A graphic spacing schedule for red pine. J. For. 1979, 77, 483–486. [Google Scholar]
- Sterba, H. Estimating potential density from thinning experiments and inventory data. For. Sci. 1987, 33, 1022–1034. [Google Scholar]
- Zeide, B. Self-thinning and stand density. For. Sci. 1991, 37, 517–523. [Google Scholar]
- Zeide, B. A relationship between size of trees and their number. For. Ecol. Manag. 1995, 72, 265–272. [Google Scholar] [CrossRef]
- Zeide, B. Natural thinning and environmental change: An ecological process model. For. Ecol. Manag. 2001, 154, 165–177. [Google Scholar] [CrossRef]
- Zeide, B. Analysis of a concept: Stand density. J. Sustain. For. 2002, 14, 51–62. [Google Scholar] [CrossRef]
- Zeide, B. Comparison of self-thinning models: An exercise in reasoning. Trees 2010, 24, 1117–1126. [Google Scholar] [CrossRef]
- Lynch, T.B.; Wittwer, R.F.; Stevenson, D.J.; Huebschmann, M.M. A maximum size-density relationship between Lorey’s mean height and trees per hectare. For. Sci. 2007, 53, 478–485. [Google Scholar]
- Inoue, A.; Nishizono, T. Allometric model of the Reineke equation for Japanese cypress (Chamaecyparis obtuse) and red pine (Pinus densiflora) stands. J. For. Res. 2004, 9, 319–324. [Google Scholar] [CrossRef]
- Woodall, C.W.; Miles, P.D.; Vissage, J.S. Determining maximum stand density index in mixed species stands for strategic-scale stocking assessments. For. Ecol. Manag. 2005, 216, 367–377. [Google Scholar] [CrossRef]
- Ducey, M.J.; Knapp, R.A. A stand density index for complex mixed species forests in the northeastern United States. For. Ecol. Manag. 2010, 260, 1613–1622. [Google Scholar] [CrossRef]
- Miyanishi, K.; Hoy, A.R.; Cavers, P.B. Generalized law of self-thinning in plant-populations. J. Theor. Biol. 1979, 78, 439–442. [Google Scholar] [CrossRef]
- Weiner, J.; Thomas, S.C. Size variability and competition in plant monocultures. Oikos 1986, 47, 211–222. [Google Scholar] [CrossRef]
- Forrester, D.I. Linking forest growth with stand structure: Tree size inequality, tree growth or resource partitioning and the asymmetry of competition. For. Ecol. Manag. 2019, 447, 139–157. [Google Scholar] [CrossRef]
- Lee, D.; Choi, J. Evaluating maximum stand density and size-density relationships based on the competition density rule in Korean pines and Japanese larch. For. Ecol. Manag. 2019, 446, 204–213. [Google Scholar] [CrossRef]
- Ge, F.W.; Zeng, W.; Ma, W.; Meng, J. Does the slope of the self-thinning line remain a constant value across different site qualities? An Implication for Plantation Density Management. Forests 2017, 8, 355. [Google Scholar] [CrossRef]
- Fu, L.H.; Zhang, J.G.; Duan, A.G.; Sun, H.G.; He, A.Y. Review of studies on maximum size-density rule. J. Plant Ecol. 2008, 32, 501–511, (In Chinese with English Abstract). [Google Scholar]
- Cheng, Z.C.; Chen, L.; Wang, G.; Zeng, S.; Fang, S. Models of Management Planning System for Pinus massoniana Forests; Chinese Forestry Press: Beijing, China, 1991. [Google Scholar]
- Cheng, Z.C.; Zeng, S. Study on Silvicultural Types and Growth and Yield Tables for Water Conservation Forests of Pinus massoniana; Chinese Forestry Press: Beijing, China, 2003. [Google Scholar]
- Fang, H.L. Evaluation of stand density index for 894 existing forests. J. Northeast For. Univ. 1995, 23, 100–105, (In Chinese with English Abstract). [Google Scholar]
- Wang, D.S. Study on stand density. For. Resour. Manag. 1994, 1, 67–71, (In Chinese with English Abstract). [Google Scholar]
- Liu, J.F.; Tong, S.Z. Growth process table of Chinese fir plantation. For. Res. 1995, 8, 164–169, (In Chinese with English Abstract). [Google Scholar]
- Liu, J.F.; Jiang, X.; Hong, W. Variable stand density yield prediction table of Pinus massoniana Lamb. Plantation in Fujian. J. Jilin For. Univ. 1999, 14, 206–209, (In Chinese with English Abstract). [Google Scholar]
- Luo, Q.B.; Zeng, W.S.; He, D.B. Model, Theory, Method and Application of Forestry Tables; Hunan Science and Technology Press: Changsha, China, 2001. [Google Scholar]
- Meng, X.Y. Forest Measurement; Chinese Forestry Press: Beijing, China, 2006. [Google Scholar]
- Che, S.H.; Zhang, J.G. Study on stand density index of Chinese fir plantation based on self-thinning rule. Bull. Bot. Res. 2012, 32, 343–347. [Google Scholar]
- Zhang, H.R. Research Method and Practice in Forest Management; Chinese Forestry Press: Beijing, China, 2018. [Google Scholar]
- Fang, J.Y.; Kan, M.; Yamakura, T. Relationships between population growth and population density in monocultures of Larix Leptolepis. Acta Bot. Sin. 1991, 33, 949–995, (In Chinese with English Abstract). [Google Scholar]
- Bi, H.Q.; Turvey, N.D. A method of selecting data points for fitting the maximum biomass- density line for stands undergoing self- thinning. Aust. Ecol. 1997, 22, 356–359. [Google Scholar] [CrossRef]
- Newton, P.E. Asymptotic size-density relationships within self-thinning black spruce and jack pine stand-types: Parameter estimation and model reformulations. For. Ecol. Manag. 2006, 226, 49–59. [Google Scholar] [CrossRef]
- Xue, L.; Ogawa, K.; Hagihara, A.; Liang, S.; Bai, J. Self-thinning exponents based on the allometric model in Chinese pine (Pinus tabulaeformis Carr.) and Prince Rupprecht’s larch (Larix principis-rupprechtii Mayr) stands. For. Ecol. Manag. 1999, 117, 87–93. [Google Scholar] [CrossRef]
- Solomon, D.S.; Zhang, L.J. Maximum size-density relationships for mixed softwoods in the northeastern USA. For. Ecol. Manag. 2002, 155, 163–170. [Google Scholar] [CrossRef]
- Huang, C.Z. stand basal area-volume tables of Schima superba Gardn. Et Champ. Plantation in Eastern Fujian. J. Fujian For. Sci. Technol. 2011, 38, 54–57, (In Chinese with English Abstract). [Google Scholar]
- VanderSchaaf, C.L. Estimating individual stand size-density trajectories and a maximum size-density relationship species boundary line slope. For. Sci. 2010, 56, 327–335. [Google Scholar]
- Xue, L.; Hagihara, A. Summary and evaluation of the researches on the self-thinning pure stands. Acta Ecol. Sin. 2001, 21, 835–838, (In Chinese with English Abstract). [Google Scholar]
- Tong, S.Z.; Sheng, W.T.; Zhang, J.G. Studies on the density effects of Chinese fir stands. For. Res. 2002, 15, 66–75, (In Chinese with English Abstract). [Google Scholar]
- Meyer, W.H. Yield of even-aged stands of ponderosa pine. USDA Tech. Bull. 1938, 630, 59. [Google Scholar]
Stand Age (Year) | Trees per Hectare (N/hm2) | Mean DBH (cm) | Mean Tree Height (m) | Basal Area (m2/hm2) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A | B | C | D | E | A | B | C | D | E | A | B | C | D | E | A | B | C | D | E | |
5 | 1667 | 3333 | 4983 | 6633 | 9967 | 5.0 | 4.8 | 4.0 | 3.8 | 3.5 | 3.7 | 3.8 | 3.5 | 3.4 | 3.3 | 3.27 | 6.03 | 6.26 | 7.52 | 9.58 |
6 | 1667 | 3333 | 4983 | 6633 | 9967 | 7.6 | 6.5 | 5.8 | 5.4 | 4.9 | 5.0 | 4.8 | 4.5 | 4.4 | 4.3 | 7.56 | 11.05 | 13.16 | 15.18 | 18.79 |
7 | 1667 | 3333 | 4983 | 6617 | 9950 | 9.8 | 8.1 | 7.0 | 6.4 | 5.8 | 6.3 | 5.9 | 5.6 | 5.2 | 5.1 | 12.57 | 17.17 | 19.17 | 21.28 | 26.28 |
8 | 1667 | 3333 | 4967 | 6583 | 9900 | 10.9 | 9.0 | 7.8 | 7.2 | 6.4 | 7.4 | 7.0 | 6.5 | 6.2 | 5.9 | 15.55 | 21.19 | 23.72 | 26.79 | 31.83 |
9 | 1667 | 3333 | 4950 | 6533 | 9733 | 12.1 | 9.9 | 8.4 | 7.8 | 6.9 | 8.4 | 7.8 | 7.2 | 6.9 | 6.7 | 19.16 | 25.64 | 27.42 | 31.20 | 36.38 |
10 | 1667 | 3317 | 4933 | 6450 | 9483 | 13.0 | 10.6 | 9.0 | 8.2 | 7.2 | 9.4 | 8.7 | 7.9 | 7.7 | 7.2 | 22.12 | 29.26 | 31.37 | 34.05 | 38.59 |
12 | 1667 | 3267 | 4883 | 6350 | 8933 | 14.4 | 11.5 | 9.5 | 9.1 | 8.0 | 10.8 | 9.9 | 9.0 | 8.8 | 8.2 | 27.14 | 33.92 | 34.59 | 41.28 | 44.88 |
14 | 1667 | 3217 | 4750 | 6050 | 8467 | 15.5 | 12.4 | 10.5 | 9.8 | 8.6 | 12.0 | 11.0 | 9.8 | 9.7 | 9.0 | 31.44 | 38.83 | 41.11 | 45.61 | 49.16 |
16 | 1633 | 3133 | 4583 | 5717 | 7467 | 16.5 | 13.1 | 11.3 | 10.3 | 9.3 | 13.0 | 11.8 | 11.0 | 10.6 | 9.9 | 34.90 | 42.21 | 45.94 | 47.61 | 50.70 |
18 | 1617 | 3067 | 4367 | 5550 | 6850 | 17.0 | 13.6 | 11.9 | 10.9 | 9.9 | 14.1 | 13.3 | 12.2 | 11.7 | 10.8 | 36.68 | 44.53 | 48.55 | 51.76 | 52.70 |
Stand Age (Year) | Trees per Hectare (N/hm2) | Mean DBH (cm) | Mean Tree Height (m) | Basal Area (m2/hm2) | Volume (m3/hm2) |
---|---|---|---|---|---|
4 | 2625 | 3.9 | 2.9 | 3.14 | 35.42 |
6 | 2790 | 7.0 | 5.3 | 10.83 | 79.75 |
8 | 2910 | 9.2 | 7.4 | 19.30 | 129.91 |
10 | 3000 | 10.6 | 9.2 | 26.32 | 181.29 |
12 | 2655 | 11.5 | 10.7 | 27.63 | 210.68 |
14 | 2655 | 12.3 | 12.1 | 31.65 | 258.61 |
16 | 2655 | 13.0 | 13.2 | 34.97 | 299.24 |
18 | 2655 | 13.44 | 14.1 | 37.67 | 332.74 |
20 | 2655 | 13.9 | 14.7 | 40.00 | 362.71 |
22 | 2640 | 14.2 | 15.2 | 41.69 | 387.43 |
24 | 2640 | 14.5 | 15.6 | 43.41 | 407.87 |
26 | 2640 | 14.7 | 16.0 | 45.05 | 428.60 |
28 | 2640 | 15.0 | 16.3 | 46.40 | 445.55 |
30 | 2640 | 15.2 | 16.5 | 47.78 | 463.17 |
Plot | Intercept (a) | Slope (b) | R2 | RMSE | RRMSE/% |
---|---|---|---|---|---|
A | −9.252 * | 3.3811 * | 0.999 | 0.29 | 1.48 |
B | −10.085 * | 4.7404 * | 0.998 | 2.61 | 10.44 |
C | −12.618 * | 5.5909 * | 0.997 | 0.48 | 2.40 |
D | −14.624 * | 6.7085 * | 0.995 | 0.67 | 3.30 |
E | −21.027 * | 9.2698 * | 0.999 | 0.04 | 0.23 |
F-line 1 | −10.471 * | 4.0123 * | 0.999 | 0.06 | 0.29 |
F-line 2 | −6.0594 * | 3.1268 * | 0.999 | 0.16 | 0.44 |
F-line 3 | −31.09 * | 4.7664 * | 0.999 | 0.03 | 0.07 |
Age | 4 | 6 | 8 | 10 | 12 | 14 | 16 |
---|---|---|---|---|---|---|---|
G′/(G − a) | 0.058 | 0.181 | 0.142 | 0.095 | 0.019 | 0.053 | 0.040 |
H′/H | 0.25 | 0.225 | 0.144 | 0.094 | 0.073 | 0.056 | 0.042 |
Age | 18 | 20 | 22 | 24 | 26 | 28 | 30 |
G′/(G − a) | 0.031 | 0.025 | 0.018 | 0.012 | 0.011 | 0.009 | 0.009 |
H′/H | 0.031 | 0.022 | 0.017 | 0.013 | 0.011 | 0.009 | 0.008 |
Age | Plot A | Plot B | Plot C | Plot D | Plot E | |||||
---|---|---|---|---|---|---|---|---|---|---|
G′/ (G − a) | H′/H | G′/ (G − a) | H′/H | G′/ (G − a) | H′/H | G′/ (G − a) | H′/H | G′/ (G − a) | H′/H | |
5 | 0.05 | 0.20 | 0.07 | 0.20 | 0.07 | 0.20 | 0.06 | 0.20 | 0.06 | 0.20 |
6 | 0.26 | 0.26 | 0.22 | 0.21 | 0.27 | 0.22 | 0.23 | 0.23 | 0.23 | 0.23 |
7 | 0.23 | 0.21 | 0.21 | 0.19 | 0.19 | 0.20 | 0.15 | 0.15 | 0.16 | 0.16 |
8 | 0.12 | 0.15 | 0.12 | 0.16 | 0.12 | 0.14 | 0.12 | 0.16 | 0.11 | 0.14 |
9 | 0.13 | 0.12 | 0.12 | 0.10 | 0.09 | 0.10 | 0.09 | 0.10 | 0.08 | 0.12 |
10 | 0.09 | 0.11 | 0.09 | 0.10 | 0.09 | 0.09 | 0.05 | 0.10 | 0.04 | 0.07 |
12 | 0.07 | 0.06 | 0.05 | 0.06 | 0.03 | 0.06 | 0.06 | 0.06 | 0.05 | 0.06 |
14 | 0.05 | 0.05 | 0.05 | 0.05 | 0.06 | 0.04 | 0.03 | 0.05 | 0.03 | 0.04 |
16 | 0.04 | 0.04 | 0.03 | 0.03 | 0.04 | 0.05 | 0.02 | 0.04 | 0.01 | 0.05 |
18 | 0.02 | 0.04 | 0.02 | 0.06 | 0.02 | 0.05 | 0.03 | 0.05 | 0.01 | 0.04 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Long, S.; Zeng, S.; Xiao, H.; Gong, Z.; Yang, S. Using Limit Value Constraint Theory to Better Understand the Self-Thinning Rule of Forest. Forests 2023, 14, 2378. https://doi.org/10.3390/f14122378
Long S, Zeng S, Xiao H, Gong Z, Yang S. Using Limit Value Constraint Theory to Better Understand the Self-Thinning Rule of Forest. Forests. 2023; 14(12):2378. https://doi.org/10.3390/f14122378
Chicago/Turabian StyleLong, Shisheng, Siqi Zeng, Huashun Xiao, Zhaosong Gong, and Shengyang Yang. 2023. "Using Limit Value Constraint Theory to Better Understand the Self-Thinning Rule of Forest" Forests 14, no. 12: 2378. https://doi.org/10.3390/f14122378
APA StyleLong, S., Zeng, S., Xiao, H., Gong, Z., & Yang, S. (2023). Using Limit Value Constraint Theory to Better Understand the Self-Thinning Rule of Forest. Forests, 14(12), 2378. https://doi.org/10.3390/f14122378