SEEFOR 15(2): 161-173
Article ID: 2416
DOI: https://doi.org/10.15177/seefor.24-16
ORIGINAL SCIENTIFIC PAPER
Predicting Diameter Distributions in Mixed Forests in Southern Mexico
Juan Carlos Guzmán-Santiago1,6, Héctor Manuel de los Santos-Posadas1, Benedicto Vargas-Larreta2, Martín Gómez-Cárdenas3,6, Wenceslao Santiago-García4, Adan Nava-Nava5*
(1) Ciencias Forestales, Colegio de Postgraduados, Campus Montecillo, Carretera México-Texcoco km 36.5, 32910 Montecillo, Texcoco, Mexico;
(2) Tecnológico Nacional de México Campus El Salto, Tecnológico 101, La Forestal, 34942 El Salto, Durango, México;
(3) Instituto Nacional de Investigaciones Forestales, Agrícolas y Pecuarias, Campo Experimental Uruapan, Avenida Latinoamericana, 47600 Michoacán, México;
(4) Universidad de la Sierra Juárez, División de Estudios de Postgrado-Instituto de Estudios Ambientales, Avenida Universidad s/n, 68725 Ixtlán de Juárez, Oaxaca, Mexico;
(5) Agropecuaria Santa Genoveva S.A.P.I. de C.V. Carretera Cayal Nohyaxche km 87, Alfredo V. Bonfil, 24570 San Francisco de Campeche, Campeche, México;
(6) Centro de Investigación, Divulgación y Asesoría Técnica Forestal y Agropecuaria (SC), Isabel la Católica, 69800 Tlaxiaco, Oaxaca, México
Citation: Guzmán-Santiago JC, de los Santos-Posadas HM, Vargas-Larreta B, Gómez-Cárdenas M, Santiago-García W, Nava-Nava A, 2024. Predicting Diameter Distributions in Mixed Forests in Southern Mexico. South-east Eur for 15(2): 161-173. https://doi.org/10.15177/seefor.24-16.
Received: 18 Aug 2024; Revised: 19 Oct 2024; Accepted: 29 Oct 2024; Published online: 6 Dec 2024
Cited by: Google Scholar
Abstract
Understanding the diameter structure of a stand is crucial for making informed decisions regarding silviculture and forest management. This is achieved by collecting forest inventory data and applying them to probability density functions. The objective of this study was to identify the optimal probability density function for describing the diameter distribution in a mixed temperate forest in Oaxaca, Mexico. Circular sampling plots of 1000 m² were established, and the diameter at breast height (DBH) and total height of 16,863 trees of the 25 most important timber species were measured. Four probability density functions (Weibull 2P, Log-Normal, Log-Logistic, and Burr) were fitted by using maximum likelihood approach for different silvicultural treatments. The predictive ability of the functions was evaluated using bias, root mean square error, Kolmogorov-Smirnov statistic and graphical analyses. In terms of accuracy and parsimony, the Burr function demonstrated the best performance, followed by the Log-Normal and Log-Logistic functions. Consequently, the Burr function is recommended for modeling the diameter distribution of stands under different treatments in the study site.
Keywords: Burr’s function; diameter distribution; goodness of fit; uneven-aged forests; silvicultural treatment
INTRODUCTION
By employing a range of silvicultural techniques, forest management enables the planning of actions that modify forest structures in accordance with pre-established objectives. In this context, diameter distribution models serve as useful tools for projecting different diameter classes present in a stand. These models facilitate the disaggregation of volume and allow for the calculation of product distribution to be extracted, as well as the potential value of the harvest (Bettinger et al. 2022). Diameter distribution analysis is directly related to forest structure, age, species composition, and site index (Bailey and Dell 1973). It also provides insight into the ecological aspects of species (Dardengo et al. 2017), and regulatory control can be implemented by assessing basal area and volume (Zhang et al. 2001, Nord-Larsen and Cao 2006). The analysis of diameter distributions provides information on the typologies and intensity of natural regeneration of species and the measurement of growth (Scolforo 2006). This, in turn, facilitates the recommendation of appropriate silvicultural treatments for different tree growth conditions (Gorgoso-Varela et al. 2020).
The use of probability density functions (PDF) in forestry has a long history, dating back to the 1970s (Bailey and Dell 1973). PDF are highly recommended for stand characterization and harvest planning (Maltamo et al. 1995, Wang and Rennolls 2005, Pérez-López et al. 2019). They provide the estimation of tree frequency by diameter category. By relating its parameters to stand attributes such as height, diameter at breast height (DBH), quadratic mean diameter, and basal area, it is possible to dynamically predict the diameter distribution (Cao 2004, Quiñonez-Barraza et al. 2015, Vega et al. 2022).
Forestry literature reports the use of PDF in silviculture for both even-aged and uneven-aged forests (Maltamo et al. 1995), which facilitate the estimation of diameter distributions. These include the Weibull, Normal, Log-Normal, Beta, Johnson’s SB, Gamma, and Log-Logistic functions. Among these, the Weibull function is the most commonly used, as evidenced by the research by Álvarez and Ruiz (1998), Fidalgo et al. (2009), and Ogana (2020).
There is a lack of such studies in Mexico, particularly in mixed forests with high diversity. This is a gap in knowledge that needs to be addressed since it hinders the proposal of functions for forests in the south of the country. The results of this study will provide valuable insights into the actual stocks, product distribution, and timber yield, significantly contributing to the sustainable management of forest resources.
The objective of this study was to compare four PDF and to identify the optimal probability density function for estimating the diameter distribution in a mixed temperate forest in southern Oaxaca, Mexico.
MATERIALS AND METHODS
Site
The study was carried out in mixed stands located in the municipality of San Juan Quiahije, in the southwest of the state of Oaxaca. According to the National Agrarian Registry (NAR), the municipality area comprises 24,914 ha. It is located at the extreme coordinates of 16°17’46” to 16°21’33” N and 97°17’54” to 97°25’53” W, with elevations ranging from 680 to 1970 m (INEGI 2023a) (Figure 1). According to environmental information from INEGI (2023b), the orography of the study site is rugged, with slopes ranging from 15% to 60%.
Figure 1.Location of the Oaxaca study site.
Data Collection with SiPlaFor
The data used in the study were obtained from the Inventory of the Forest Management Unit number 2012, where the Mexican Method Management for Uneven-aged Forests is applied. A total of 450 circular sampling plots, each of 1,000 m2 size, were established using randomized (block) design (SiPlaFor 2015), covering an area of 3,043 ha. The sample size comprises 16,863 trees, representing 25 tree species. During the field campaign, on each plot tree height was measured with a Haga® hypsometer (Haga GmbH and Co. KG, Nuremberg, Germany) (Cho et al. 2009), and the tree DBH for trees with minimal DBH of 5 cm was measured with a diameter tape Model 283D.
The largest measured tree dimensions belong to Pinus maximinoi, P. douglasiana and P. oocarpa, which exceeded 100 cm in DBH and 35 m in height, confirming them as dominant species. Most species have leptokurtic diameter distribution, while some such as Quercus candicans and Q. castanea, have platykurtic distribution. The positive asymmetry in diameter distribution was found for all species, except for Pinus ayacahuite, showing the negative skew (Table 1).
Table 1. Descriptive statistics of the main tree species used in the analyses.
Silvicultural Treatments in Stands
Silviculture consists of treatments such as regeneration cuts and intermediate cutting, which are essential for managing the diameter structure of forest species (Daniel et al. 1982, Nyland 2002). In the site area, the Mexican Method Management for Uneven-aged Forests was used, aimed at applying selective cutting to promote the maintenance of an uneven-aged structure composed of trees of different dimensions (Hernández-Díaz et al. 2008, Torres-Rojo et al. 2016). Given the presence of uneven-aged stands in the site area, silvicultural work has been implemented, including thinning activities (1st thinning, 2nd thinning, and 3rd thinning), which involve eliminating poor quality individuals without affecting the entire stand and thus favoring the most vigorous trees. Additionally, regeneration cutting (RGC) has been implemented to support thinning, eliminating competing trees that surround the selected ones, in addition to complementing with release cutting (RC) that allows light to enter the selected areas of the canopy and facilitates the natural regeneration of the species (Hernández-Díaz et al. 2008).
There are also unavailable forests (ND, segregated) intended to conserve and thus maintain ecological functions (biodiversity). Group selection cutting (SELG) is another important treatment. This method maintains the forest structure (age and size) by removing groups of tree species in a controlled manner, creating space for natural regeneration (Puettmann et al. 2016, Torres-Rojo et al. 2016).
In all treatments, the most dominant species were Pinus maximinoi and P. oocarpa. Those species were even more predominant in areas not segregated (ND) by moderate forest harvesting and in group selection cutting (SELG), where an average DBH of 23 cm was recorded. However, the SELG cutting resulted with a greater range of diameter class, ranging from 6 to 125 cm. In this context, the majority of the forest stands are dominated by conifers across each of the silvicultural treatments, which was fundamental for the fitting and analysis of the functions (Table 2).
Probability Density Functions (PDF)
In the diameter distribution analysis, four PDF recommended in the studied literature (de Lima et al. 2017, Gorgoso-Varela et al. 2020, Ogana 2020, de Lima et al. 2023, Gorgoso-Varela et al. 2024) were fitted to describe the diameter distribution of the forest stands (Table 3).
Table 3. Probability density functions (PDF) fitted to the data.
Method for Fitting and Selection of the Functions
The maximum likelihood (MLE) method was used to estimate the parameters of each PDF using the fitdistrplus (Müller and Dutang 2015) and MASS (Venables and Ripley 2002) packages in the Rstudio programming environment (R Development Core Team 2023). The MLE approach consists of estimating the parameters by maximizing the probability of obtaining a sample from the theoretical distribution that matches the observed distribution (Bailey and Dell 1973, Ciceu et al. 2021, Gorgoso-Varela et al. 2021).
The analysis of the metrics to select the best function consisted of numerical and graphical comparisons of the bias (Bias, Equation 1) and the root mean squared error (RMSE, Equation 2), where it is expected that the values approach zero.
(1)
(2)
where F(xi) is the number of observed trees in diameter class i, F^(xi) is the number of estimated trees in diameter class i, n is the number of observations and p is the number of parameters of the PDF.
Bias and RMSE values were estimated for each PDF as the average of the relative frequencies of each diameter class (at 5 cm intervals) for each forest inventory plot, stand, and silvicultural treatment. The Kolmogorov-Smirnov statistic (Dn) was used to validate the similarity between the empirical cumulative Fn(xi) and theoretical F0(xj) frequencies: Dn:supx|Fn(x)-F0(x)|, where supx is the maximum of the sum of the set of distances.
This value was calculated as follows (Equation 3, Cao 2004):
(3)
The following expression was used to compare the obtained value (Dn) with the critical value proposed by Miller (1956); the probability error was determined to be 0.05 (Equation 4, Gorgoso-Varela et al. 2018):
(4)
where ln is the natural logarithm, α is the significance level, and n is the number of samples considered in each silvicultural treatment. If Dn > Dn,α, the null hypothesis that the data distribution follows a theoretical distribution is rejected (Gorgoso-Varela et al. 2018).
RESULTS AND DISCUSSION
Evaluation of Probability Density Functions (PDF)
Table 4 shows the most commonly used goodness-of-fit metrics for each PDF by silvicultural treatment with a 95% confidence level. Reliable values were observed for both the bias and RMSE, with biases being consistently below zero.
The Burr function (see Table 4 and Figure 3) is considered the most suitable for describing the diameter distribution of the stands under different silvicultural treatments, followed by the Log-Normal and Log-Logistic functions (Table 4). The estimated parameters for each function and treatment are shown in Table A1 (Appendix A). While the Burr function is preferred, the Log-Normal and Log-Logistic functions are also viable alternatives, with the Log-Logistic slightly outperforming the Log-Normal in projections. The choice of the Burr PDF is supported by Sahin and Ercanli (2023), who found higher fit values among 14 PDF for northern Turkey forests, noting RMSE values between 27.60 and 71.06, even when using the same functions such as Weibull 2P and Log-Normal.
Similarly, Siipilehto and Mehtätalo (2013), Diamantopoulou et al. (2015) and Schmidt et al. (2020) reported significant modeling errors, highlighting the challenges in accurately modeling diameter distributions in forestry.
Diameter Distribution with PDF
Diameter distribution is important for analyzing forest growth since it allows understanding the structure and dynamics of the tree population within an ecosystem through forest inventory. It provides detailed information on tree density (de Lima et al. 2023) and helps diagnose mortality, regeneration, and other events affecting the forest stand (Imaña-Encinas et al. 2009).
Our findings confirm the situation that most trees are grouped in the 5 to 30 cm diameter range (Figure 2), indicating effective regeneration due to forest management activities. Appropriate decision-making tools, such as PDF, are essential in this context. The Burr function was found to accurately describe all diameter classes for each treatment. However, it slightly overestimated diameter classes from 10 to 20 cm for the 1Thin and SELG treatments (Figure 3). While there have been studies on diameter distribution in Mexico (Corral-Rivas et al. 2015, Quiñonez-Barraza et al. 2015), southern Mexican forests are still not covered enough by this type of research (Pérez-López et al. 2019, Nava-Nava et al. 2024).
The Log-Logistic and Log-Normal PDF performed well in modeling the data, particularly for the 1Thin, RGC, ND, and SELG treatments. The Log-Normal function has also shown significant results in other studies (de Brito et al. 2022). However, Maltamo (1997) suggests that species-fitted functions often provide even better results.
The Weibull function, in its various forms, is known as flexible and has achieved significant results in diverse forest conditions (Reynolds et al. 1998, Liu et al. 2004, Lei 2008, Corral-Rivas et al. 2015, Pérez-López et al. 2019, Pogoda et al. 2019, Nava-Nava et al. 2024). Considering our research, the Weibull function did not perform as expected. Specifically, the two-parameter Weibull function tended to underestimate the measured values, particular in diameter classes from 20 to 30 cm. Sun et al. (2019) suggest that increasing sample area and plot size could improve PDF accuracy in even-aged stands. In contrast, Machado et al. (2009), Canuto-Amaral (2017), Medeiros et al. (2018), and de Lima et al. (2023) found that the two-parameter Weibull function performed poorly predicting diameter frequencies.
Uneven-aged natural forests often show an “inverted J” shape of diameter distribution, a characteristic of geometric series, as documented by Liocourt (1898) and de Lima et al. (2023). This unimodal distribution is positively skewed (de Lima et al. 2017, 2023). However, each species has a specific growth pattern and possesses unique adaptive traits, particularly during the juvenile stage, when rapid growth occurs.
The results show that Pinus maximinoi and P. oocarpa are the dominant species on all sampling plots, highlighting their ecological significance in southern Oaxaca. As pointed out by de Brito et al. (2022), when there is a concentration of young trees in a stand, there is a high probability that they will reach the adult stage, which guarantees the long-term sustainability of the species in the region.
Statistical Evaluation of PDF
We evaluated model performance by comparing the estimated inaccuracies in diameter class. Assessing bias and RMSE is essential for understanding the accuracy and precision of predictions (Guzmán-Santiago et al. 2020). As shown in Figures 4 and 5, the Burr function had the lowest estimation errors for both bias and RMSE. However, it should be noted that the model tends to slightly underestimate trees with DBH ranging between 5 and 15 cm. These results correspond with findings from other studies (Vega et al. 2022, Sahin and Ercanli 2023), which identified the largest bias in the smallest diameter classes (5-15 cm). This is often related to silvicultural interventions and competition among young trees, as noted by Corral-Rivas et al. (2015) in conifer and broadleaf tree species in Durango, Mexico.
With the impact of these results on the region, a new base knowledge has been created for further studies on dynamic estimation modeling of diameter distribution and tree volume over time.
CONCLUSIONS
The Burr function proved to be the most suitable for describing diameter distribution in mixed stands across the seven silvicultural treatments evaluated in southern Mexico. The function is recommended to serve as an effective tool for describing the diameter distribution of stands relative to silvicultural prescriptions, supporting forest management planning and forest products distribution in the region.
Author Contributions
JCGS and ANN designed the research, performed the statistical analysis, and drafted the manuscript, HMSP, BVL, MGC, and WSG reviewed the writing. All authors read and approved the final manuscript.
Funding
This research received no external funding.
Acknowledgments
The first author is grateful to the Consejo Nacional de Humanidades, Ciencia y Tecnología (CONAHCYT) for granting the scholarship for the Postdoctoral stay at the Colegio de Postgraduados en Ciencias Forestales, Campus Montecillo, Texcoco, Mexico; as well as to the agrarian authorities of the community of San Juan Quiahije, Juquila, together with the forest technicians (Ing. Gregorio Morales and Bio. María del Sol), for the support in the field measurements.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A
Table A1. Estimated parameters of the probability distribution functions by silvicultural treatment.
REFERENCES
Álvarez JG, Ruiz AD, 1998. Analysis and modeling of diameter distributions of Pinus pinaster Ait. in Galicia. Inv Agrar Sist Recur For 7(1-2): 123-137. https://doi.org/10.5424/595.
Bailey RL, Dell TR, 1973. Quantifying diameter distributions with the Weibull function. For Sci 19(2): 97- 104. https://doi.org/10.1093/forestscience/19.2.97.
Bettinger P, Izlar B, Harris T, Cieszewski C, Conrad J, Greene D, et al. 2022. Handbook of land and tree measurements. University of Georgia, Harley Langdale, Jr. Center for Forest Business, Athens, Georgia, USA, 325 p.
Canuto-Amaral G, Medeiros-Lustosa IJr, Karolina-Barros de Oliveira E, Canuto-Amaral FH, Brito de Azevedo G, Oliveira-Castro RV, et al., 2017. Diameter distribution of Cerrado – Caatinga forest formations using different Weibull functions forms. Aust J Basic Appl Sci 11(9): 1-6.
Cao Q, 2004. Predicting parameters of a Weibull function for modeling diameter distribution. For Sci 50: 682-685. https://doi.org/10.1093/forestscience/50.5.682.
Cho MA, Skidmore AK, Sobhan I, 2009. Mapping beech (Fagus sylvatica L.) forest structure with airborne hyperspectral imagery. Int J Appl Earth Obs Geoinf 11: 201-211.
Ciceu A, Pitar D, Badea O, 2021. Modeling the diameter distribution of mixed uneven-aged stands in the South Western Carpathians in Romania. Forests 12(7): 958. https://doi.org/10.3390/f12070958.
Corral-Rivas S, Álvarez-González JG, Corral-Rivas JJ, López-Sánchez CA, 2015. Characterization of diameter structures of natural forests of northwest of Durango, Mexico. Rev Chapingo Ser Cienc For Ambient 21(2): 221-236. https://doi.org/10.5154/r.rchscfa.2014.10.046.
Daniel PW, Helms UE, Baker FS, 1982. Principios de Silvicultura. McGraw Hill, México, D.F. 492 p.
Dardengo JDFE, Rossi AAB, da Silva CJ, Silveira M, 2017. Spatial structure of Theobroma subincanum Mart. and Theobroma speciosum Willd. ex Spreng. in the Parque Nacional Do Juruena, Mato Grosso state, Brazil. Rev Árvore 41(1): e410101. https://doi.org/10.1590/1806-90882017000100001.
de Brito AC, de Paula A, Barreto-Garcia PAB, de Lima RB, Souza DR, 2022. Diametric distribution modeling of forest species in arboreal Caatinga vegetation, Brazil. Rev Ceres 69: 464-469. https://doi.org/10.1590/0034-737X202269040011.
de Lima RB, Bufalino L, Alves Junior FT, de Silva JAA, Ferreira RLC, 2017. Diameter distribution in a Brazilian tropical dry forest domain: predictions for the stand and species. An Acad Bras Ciênc 89(2): 1189-1203. https://doi.org/10.1590/0001-3765201720160331.
de Lima RB, dos Santos-Neto JC, de Brito AC, de Oliveira CP, de Paula A, Barreto-Garcia PAB, 2023. Quantifying diameter distributions in seasonally dry tropical forest in Bahia, Brazil. Rev Bras Ciênc Agrár 18(1): e2992. https://doi.org/10.5039/agraria.v18i1a2992.
Diamantopoulou MJ, Özçelik R, Crecente-Campo F, Eler Ü, 2015. Estimation of Weibull function parameters for modelling tree diameter distribution using least squares and artificial neural networks methods. Biosyst Eng 133: 33-45. https://doi.org/10.1016/j.biosystemseng.2015.02.013.
Fidalgo T, Pacheco C, Parresol BR, 2009. Describing maritime pine diameter distribution with Jhonson’s SB distribution using a new all-parameter recovery approach. For Sci 55(4): 367-373. https://doi.org/10.1093/forestscience/55.4.367.
Gorgoso-Varela JJ, Adedapo SM, Ogana FN, 2024. A comparison of probability density functions fitted by moments and maximum likelihood estimation methods used for diameter distribution estimation. Forests 15(3): 425. https://doi.org/10.3390/f15030425.
Gorgoso-Varela JJ, Ogana FN, Ige PO, 2020. A comparison between derivative and numerical optimization methods used for diameter distribution estimation. Scand J For Res 35(3-4): 1-9. https://doi.org/10.1080/02827581.2020.1760343.
Gorgoso-Varela JJ, Oliveira-Prendes JA, Afif-Khouri E, Palencia P, 2018. The weibull distribution to describe aboveground biomass of Lolium multiflorum Lam., in relation to rate of nitrogen application. Agrociencia 52(1): 67-75.
Gorgoso-Varela JJ, Ponce RA, Rodríguez-Puerta F, 2021. Modeling diameter distributions with six probability density functions in Pinus halepensis Mill. plantations using low-density airborne laser scanning data in Aragón (Northeast Spain). Remote Sens 13(12): 2307. https://doi.org/10.3390/rs13122307.
Guzmán-Santiago JC, Aguirre-Calderón OA, Jiménez-Pérez J, Vargas-Larreta B, 2020. Volume estimation of Abies religiosa (Kunth) Schltdl. & Cham. In different states of Mexico. Colomb For 23(2): 99-113. https://doi.org/10.14483/2256201X.15557.
Hernández-Díaz JC, Corral-Rivas JJ, Quiñones-Chávez A, Bacon-Sobbe JR, Vargas-Larreta B, 2008. Evaluación del manejo forestal regular e irregular en bosques de la Sierra Madre Occidental. Madera Bosques 14(3): 25-41.
Imaña-Encinas J, Santana OA, Ribeiro GS, Imanã CR, 2009. Diameter structure of one fragment of the semi-deciduous seasonal forest in Cerrado Ecomuseum Area, Goiás. Cerne 15(2): 155-165.
INEGI, 2023a. División por entidad federativa con base en el marco geoestadístico, Instituto Nacional de Estadística y Geografía. Ciudad de México. Available online: www.inegi.gob.mx/geo/informaciongeografica/oaxaca (15 October 2023).
INEGI, 2023b. Sistemas de consulta. Espacio y datos de México. Instituto Nacional de Estadística y Geografía. Ciudad de México. Available online: www.inegi.org.mx/app/mapa/espacioydatos/default.aspx?ag=202130001 (15 October 2023).
Lei Y, 2008. Evaluation of three methods for estimating the Weibull distribution parameters of Chinese pine (Pinus tabulaeformis). J For Sci 54(12): 566-571. https://doi.org/10.17221/68/2008-JFS.
Liocourt F, 1898. De l’amenagement des sapinières. Bulletin Trimestriel - Société Forestière de Franche-Comté et Belfort 4 (6): 396-409. Available online: https://infodoc.agroparistech.fr/index.php?lvl=notice_display&id=181924 (5 October 2023).
Liu A, Li F, Zhang L, Jin X, 2014. Modeling diameter distributions of mixed species forest stands. Scand J For Res 29(7): 653-663. https://doi.org/10.1080/02827581 .2014.960891.
Machado SA, Augustynczik ALD, Nascimento RGM, Téo SJ, Miguel EP, Figura MA, Silva LCR, 2009. Diametric distribution functions in a fragment of Mixed Ombrophylous Forest. Ciênc Rural 39(8): 2428-2434. https://doi.org/10.1590/S0103-84782009000800024.
Maltamo M, 1997. Comparing basal area diameter distributions estimated by tree species and for the entire growing stock in a mixed stand. Silva fenn 31(1): 5609. https://doi.org/10.14214/sf.a8510.
Maltamo M, Puumalainen J, Päivinen R, 1995. Comparison of Beta and Weibull functions for modeling basal area diameter distribution in stands of Pinus sylvestris and Picea abies. Scand J Forest Res 10(1-4): 284-295. https://doi.org/10.1080/02827589509382895.
Medeiros FS, Souza MP, Cerqueira CL, Alves AR, Souza MS, Borges CHA, 2018. Florística, fitossociologia e modelagem da distribuição diamétrica em um fragmento de Caatinga em São Mamede-PB. Agropecuária Científica no Semiárido 14: 85-95. https://doi.org/10.30969/acsa.v14i2.900.
Miller LH, 1956. Tables of percentage points of Kolmogorov statistic. J Amer Statist Assoc 51: 111-121. https://doi.org/10.1080/01621459.1956.10501314.
Müller MLD, Dutang C, 2015. fitdistrplus: an R package for fitting distributions. J Statist Software 64(4): 1-34. https://doi.org/10.18637/jss.v064.i04.
Nava-Nava A, Ángeles-Pérez G, Santiago-García W, Quiñonez-Barraza G, de los Santos-Posadas HM, Valdez-Lazalde JR, 2024. Bayesian approach for modeling diameter distributions of patula pine (Pinus patula Schiede ex Schltdl. & Cham.) stands in Southern Mexico. Trees For People 16: 100567. https://doi.org/10.1016/j.tfp.2024.100567.
Nord-Larsen T, Cao QV, 2006. A diameter distribution model for even-aged beech in Denmark. Forest Ecol Manag 231(1-3): 218-225. https://doi.org/10.1016/j.foreco.2006.05.054.
Nyland RD, 2002. Silviculture. Concepts and applicactions. Waveland Press, Inc., Illinois, USA. 682 p.
Ogana FN, 2020. A percentile-based estimator for the log-logistic function: Application to forestry. Forestry Studies 72(1): 107-120. https://doi.org/10.2478/fsmu-2020-0009.
Pérez-López E, Santiago-García W, Quiñonez-Barraza G, Rodríguez-Ortiz G, Santiago-García E, Ruiz-Aquino F, 2019. Estimation of diameter distributions for Pinus patula with the Weibull function. Madera Bosques 25(3): e2531626. https://doi.org/10.21829/myb.2019.2531626.
Pogoda P, Ochal W, Orzel S, 2019. Modeling diameter distribution of Black Alder (Alnus glutinosa (L.) Gaertn.) stands in Poland. Forests 10(5): 412. https://doi.org/10.3390/f10050412.
Puettmann KJ, Coates K, Messier C, 2016. Crítica de la Silvicultura el manejo de la complejidad. Editorial: ACCI ediciones - Asociación cultural y científica iberoamericana, 294 p.
Quiñonez-Barraza G, De los Santos-Posadas HM, Cruz-Cobos F, Velázquez-Martínez A, Ángeles-Pérez G, Ramírez-Valverde G, 2015. Dynamic modeling for diameter distribution on Pinus mixed stands in Durango, Mexico. Madera Bosques 21(2): 59-71.
R Development Core Team (2022) R. A Language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Available online: http://www.rstudio.com/ (15 October 2023).
Reynolds MR, Burk TE, Huang WC, 1988. Goodness-offit tests and model selection procedures for diameter distribution models. For Sci 34: 373-399. https://doi.org/10.1093/forestscience/34.2.373.
Sahin A, Ercanli I, 2023. An evaluation of various probability density functions for predicting diameter distributions in pure and mixed-species stands in Türkiye. For Syst 32(3): e016. https://doi.org/10.5424/fs/2023323-20130.
Schmidt LN, Sanquetta MNI, McTague JP, Da Silva GF, Filho CVF, Sanquetta CR, et al., 2020. On the use of Weibull distribution in modelling and describing diameter distributions of clonal eucalypt stands. Can J For Res 50(10): 1050-1063. https://doi.org/10.1139/cjfr2020-0051.
Scolforo JRS. 2006. Biometria florestal: modelos de crescimento e produção florestal. Lavras: UFLA/ FAEPE.
Siipilehto J, Mehtätalo L, 2013. Parameter recovery vs. parameter prediction for the Weibull distribution validated for Scots pine stands in Finland. Silv Fenn 47(4): 1-22. https://doi.org/10.14214/sf.1057.
SiPlaFor, Sistema de Planeación Forestal de Bosque Templado (2015) Manual de usuario. Versión 2.0, Comisión Nacional Forestal. Zapopan, Jalisco, México. Available online: http://siplafor.cnf.gob.mx/siplafor/inicio/index.php (15 October 2023).
Smith DM, Larson BC, Kelty MJ, Ashton PMS, 1997. The practice of silviculture. Applied forest ecology. John Wiley and Sons, New York, USA. 537 p.
Sun S, Cao QV, Cao T, 2019. Characterizing diameter distributions for uneven-aged pine-oak mixed forests in the Qinling Mountains of China. Forests 10(7): 596. https://doi.org/10.3390/f10070596.
Torres-Rojo JM, Moreno-Sánchez R, Mendoza-Briseño MA, 2016. Sustainable Forest Management in Mexico. Curr Forestry Rep 2: 93-105. https://doi.org/10.1007/s40725-016-0033-0.
Vega AA, Corral-Rivas S, Corral-Rivas JJ, Diéguez-Aranda U, 2022. Modelación de las estructuras diamétricas en bosques naturales de Pueblo Nuevo, Durango. Rev Mex Cienc For 13(73): 75-101. https://doi.org/10.29298/rmcf.v13i73.1187.
Venables WN, Ripley BD, 2002. Modern Applied Statistics with S. Fourth Edition. Springer, New York, USA. 498 p. https://doi.org/10.1007/978-0-387-21706-2.
Wang M, Rennolls K, 2005. Tree diameter distribution modeling: Introducing the logit–logistic distribution. Can J For Res 35(6): 1305–1313. https://doi.org/10.1139/x05-057.
Zhang L, Gove JH, Liu Ch, Leak WB, 2001. A finite of two Weibull distributions for modeling the diameter distributions of rotated-sigmoid, uneven-aged stands. Can J For Res 31(9): 1654-1659. https://doi.org/10.1139/x01-086.
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