Original article

Equation to predict the maximum crown diameter of Pinus cooperi Blanco in Durango, México

0000-0002-6992-3511José Guadalupe Colín¹  , 0000-0001-5668-8869Oscar Alberto Aguirre Calderón²  , 0000-0002-2851-7517José Javier Corral Rivas³  , 0000-0002-6106-007XTilo Gustavo Domínguez Gómez¹  , 0000-0003-0480-624XFrancisco Javier Hernández¹  , 0000-0001-5133-0403Erik Orlando Luna Robles^{1  *} 

¹Instituto Tecnológico de El Salto. México.

²Universidad Autónoma de Nuevo León. Facultad de Ciencias Forestales. México.

³Universidad Juárez del Estado de Durango. Facultad de Ciencias Forestales y Ambientales de la México. México.

ABSTRACT

The maximum crown area is an important variable in the estimation of the level of competition that affects the growth potential and productivity of species. Therefore, the objective of the present study was to adjust regression models to predict the maximum crown diameter for Pinus cooperi Blanco in the forest region of El Salto, in the state of Durango. In particular, 95 trees of different diameter categories and growing free of competition were used. Independently, three regression models were analyzed, simple linear, second-degree polynomial and potential, to predict the maximum canopy diameter as a function of normal diameter, total height, height at the base of the crown, age and total length of the crown. The results showed that the second-degree polynomial model with the inclusion of normal diameter as an independent variable presented the best fit in the prediction of the maximum crown diameter. In general, normal diameter was the most significant variable for the prediction of maximum crown diameter in Pinus cooperi. Based on the results, two forest productivity scenarios related to crown competition factor (CCF) can be identified; when the CCF is 100 % or higher, timber productivity is optimal and when it is less than 100 %, productivity is not optimal, which can be attributed to little or no competition and deficiency in land use..

Key words: maximum area; competition; models; timber productivity.

INTRODUCTION

Timber production depends on the level of productivity existing within a given forest area. In the period from 1990 to 2017, the total national timber production was, on average, 6.86 million m³ of roundwood, and pine contributed 79.38 % (^{SEMARNAT, 2019}), but the effective use of the land and the quality of the forest products produced there depend on the management of the density of the stands throughout the rotation, in this sense, management can be considered as a fundamental quantitative tool for planning, execution and evaluation that defines silvicultural interventions (^{Santiago-García et al ., 2013}; ^{Vospernik and Sterba, 2015}; ^{Tamarit et al., 2020}). In addition to the above, forest development depends on the different morphological attributes of the trees such as the shape of the trunks, roots and crowns, components that are usually evaluated to predict the growth and productivity of forests. However, there are few studies that consider crown parameters (Nájera and Hernández, 2008), despite the fact that the variables related to tree crowns offer interdimensional information such as the surface area occupied by an individual, level of competence and vitality (^{Hess et al., 2016}; ^{Cisneros et al., 2019}; ^{Givnish, 2020}). Therefore, forest management defines the morphometric relationships in forest communities, so their adequate description and characterization can help in the evaluation of silvicultural practices (^{Soto et al., 2016}).

The size, structure and shape of tree canopies determine the extent and efficiency of physiological processes such as photosynthetic activity, transpiration and respiration which in turn determine the growth, development and productivity of the forest (^{Sharma et al., 2017}; ^{Cabon et al., 2018}; ^{Hernández et al., 2022}; ^{Sporek and Sporek, 2023}). Analysis, monitoring and modeling of ecosystems with more precise, sophisticated and detailed techniques that demand less work to directly measure crown diameter (^{Pretzsch, 2022}). Evaluating the potential density of the target population stand helps to apply the appropriate silvicultural treatments (^{Yang and Brandeis, 2022}). In this way, the timely management of the density of these areas can be decisive in accelerating the growth of residual trees, which is why mathematical tools are required that relate the size of the trees with their number. In recent decades, different methodologies have been developed to determine the density level of a stand, such as the ^{Reineke density index (Reineke, 1933}; ^{Curtis and Reukema, 1970}), which is based on the maximum density that a stand can support, and the crown competition factor (CCF) (^{Krajicek et al., 1961}), which helps to determine the number of trees of each diameter category that a hectare can support just at the threshold of the beginning of competition and is estimated from the determination of the maximum crown area projected by trees growing free of competition.

The maximum crown projection area is an important measure for the development of density guides (^{Rodríguez et al., 2009}; ^{Hernández et al., 2013}; ^{Bueno et al., 2022})), for the projection of the growth of individual trees as a function of density (^{Biging and Dobbertin, 1995}; ^{Hasenauer et al., 1994}; ^{Pretzsch et al., 2022}), ^{Pretzsch et al., 2002}) for the determination of light conditions in the understory that in turn are important for the establishment and development of regeneration (^{Crookston and Stage , 1999}), and for the evaluation of the effect of competition on individual trees (^{Smith et al., 1992}; ^{Corral et al ., 2004}; ^{Colin et al., 2018}; ^{Arnoni et al., 2020}). Indeed, crown characteristics are also sometimes used to attribute social ranks to trees in a stand, driving the marking of trees for thinning (^{Bravo et al., 2020}).

For the development of maximum crown diameter models, the crown diameter of a sample of trees growing in open spaces free of competition is usually related to their normal diameter (^{Bechtold, 2003}; ^{Yang and Huang, 2017}: Qiu et al., 2023). Other variables such as the geographic location of the trees, elevation, exposure and slope have provided marginal improvements in the estimation of the maximum crown diameter (^{Paine and Hann, 1982}; ^{Hasenauer, 1997}). Currently, the use of indicators and statistical models to support forest management practices is increasing (^{Marchi et al., 2020}). Despite the importance of the crown competition factor in the management of forest natural resources, there are currently no equations to estimate the maximum crown size of most commercially important forest species in the state of Durango. For this reason, the objective of this study was to compare different regression equations to predict the maximum crown diameter of Pinus cooperi in the forest region of El Salto, Durango.

MATERIALS AND METHODS

Area of study

The study was carried out in the Regional Forest Management Unit 1008 (UMAFOR 1008), which includes the municipality of Pueblo Nuevo and part of the municipality of Durango. The UMAFOR covers approximately 507,127 ha and is located in the mountainous massif of the Sierra Madre Occidental, southwest of the state of Durango (Figure 1). The predominant type of vegetation corresponds to mixed forests with species of the genera Pinus and Quercus mainly. The height above sea level varies from 2,400 m to 2,600 m. The prevailing climate is temperate semi-cold with an annual precipitation regime that fluctuates from 900 to 1,200 mm and an average annual temperature that varies from 8°C in the highest parts to 24°C in the lowest parts (^{Instituto Nacional de Estadística Geografía e Informática [INEGI], 2015}).

Fig. 1.  - Location of sampling plots of Pinus cooperi 

Sampling

The data come from 95 trees randomly selected through targeted sampling. The main characteristic for selecting the sample tree was that it would not be found growing in competition with other trees, nor that there were any stumps close to the target tree within a 30-meter radius, in order to ensure that it developed in a competition-free environment. For each selected tree, the normal diameter (D), total height (HT), clean stem height (ABLC) (insertion of the first whorl with live needles that is part of the crown as a whole), height of the first live needles (ABLF), age in years, crown radius in four directions that coincide with the cardinal points (CR), crown length (CL) were recorded (Figure 2).

Fig. 2.  - Main crown variables and geometric relationships used in the development of the maximum crown diameter equation. HBLC: height of the crown base (m); HBLF: height of the first living needles (m); CR: radius of the crown at each measurement point (m); CH: height of the crown from HBLC (m); CL: total length of the crown (m); HT: total height of the tree (m) 

Likewise, the name of the property, UTM Datum WGS 84 coordinates, height above sea level, exposure and slope were obtained as control information for each tree. Table 1 presents the most important descriptive statistics for the 95 sample trees.

Table 1.  - Descriptive statistics of the sample trees 

Variable	Average	Maximum	Minimum	Standard deviation
D	44.97	83.5	23.50	14.56
HT	10.75	20.4	5.60	3.87
HBLC	2.55	8.8	0.07	1.89
HBLF	1.19	4.8	0.07	0.99
HCM	4.43	12.0	0.2	2.44
CL	8.21	14.3	4.2	2.43
DMC	8.99	15.1	4.78	2.58

D: normal diameter (cm), HT: total height (m), HBLC: height of the base of the crown (insertion of the first whorl with living needles that is part of the crown as a whole) (m), HBLF: height of the first living needles (m); HCM: maximum crown height, CL: total crown length (m), and DMC: maximum crown diameter.

Models

Three regression models (simple linear, second-degree polynomial and potential) were fitted to predict maximum crown diameter independently for each of the predictor variables D, HT, HBLC and age using the ordinary least squares (OLS) technique, with the MODEL procedure of the SAS/ETS® program (SAS Institute Inc., 2008). The mathematical expressions of the models analyzed are Equation 1, Equation 2 and Equation 3:

$dmc={\beta }_0+{\beta }_1*Vi$ (8)

$dmc={\beta }_0+{\beta }_1*Vi+{\beta }_2*{Vi}^2$ (9)

$dmc={\beta }_0*{Vi}^{{\beta }_1}$ (10)

Where: $dmc:$ maximum crown diameter; $Vi$ : Independent variable; $\beta i$ : are the parameters to be estimated in the adjustment.

Using the best model as a base, the maximum canopy area and percentage of coverage by diameter category were estimated, as well as the number of trees and the basal area sufficient to cover one hectare. The relationship between the number of trees and the average normal diameter allowed the elaboration of a density graph at different percentages of canopy coverage.

Model comparison and selection

The analysis of the adjustment capacity of the equations was based on the graphical analysis of the residues and on the values of two statistics: the coefficient of determination (R²) and the root mean square error (RMSE), whose mathematical expressions are the following Equation 4 and Equation 5:

${\mathrm{R}}^2=1-\mathrm{ }\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{n}}{\left({\mathrm{Y}}_{\mathrm{i}}-{\widehat{\mathrm{Y}}}_{\mathrm{i}}\right)}^2}{{\sum }_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{n}}{\left({\mathrm{Y}}_{\mathrm{i}}-\stackrel{-}{\mathrm{Y}}\right)}^2}$ (11)

$\mathrm{R}\mathrm{E}\mathrm{M}\mathrm{C}=\sqrt{\frac{\sum _{\mathrm{i}=1}^{\mathrm{i}=\mathrm{n}}{\left({\mathrm{Y}}_{\mathrm{i}}-{\widehat{\mathrm{Y}}}_{\mathrm{i}}\right)}^2}{\mathrm{n}-\mathrm{p}}}$ (12)

Where: Y, Ŷ, $\stackrel{-}{\mathrm{Y}}$ are respectively the observed, estimated and average values of the dependent variable, n is the total number of observations used to fit the model and p is the number of parameters to estimate.

RESULTS AND DISCUSSION

The statistical analysis indicate that the independent variable that presented the best fit in the three models used to estimate the maximum crown diameter of Pinus cooperi was the normal diameter (R² from 0.81 to 0.83 and RCME from 1.07 to 1.10), followed by total height and age (Table 2). Similar results were obtained by ^{Coombes (2019}) where he obtained a higher adjusted R² (0.85) using quadratic regression. Crown diameter estimates typically employ regression models that use normal diameter as their main explanatory variable due to its high correlation with crown diameter and easy measurement compared to other variables such as crown height and length, among others (^{Quadri, 2019}). Studies carried out on a variety of wood species have shown that normal diameter has been a reliable predictor variable in estimating maximum crown diameter (^{Bechtold, 2003}; ^{Condes and Sterba, 2005}; ^{Rodríguez et al., 2009}; ^{Martin et al., 2012}; ^{Chen et al., 2021}; ^{Qiu et al., 2022}; Qiu et al., 2023). Although other authors have used normal diameter as the main predictor variable, total height has been added to this variable (^{Li, 2020}; Pretzsch et al., 2020), crown length (^{Moeur, 1981}), crown area projected above ground (^{Jucker et al., 2017}; ^{González- Benecke et al., 2022}), density (^{Bragg, 2001}; ^{Sporek and Sporek , 2023}) exposure, slope and altitude (^{Curtis and Reukema 1970}) and in the use of artificial neural networks to improve the level of prediction (^{Bueno et al., 2022}; ^{Ou and Quiñonez, 2023}).

Table 2.  - Fit statistics of the variables to model the maximum crown diameter of Pinus cooperi  

Model	Predictor variable
	DN		HT		HBLC		EDAD
	R^2	REMC	R^2	REMC	R^2	REMC	R^2	REMC
$\left[1\right]$	0,81	1,10	0,57	1,70	0,27	2,21	0,50	1,82
$\left[2\right]$	0,83	1,07	0,66	1,50	0,38	2,05	0,55	1,73
$\left[3\right]$	0,82	1,09	0,59	1,64	0,33	2,11	0,54	1,75

DN: normal diameter (cm), HT: total height (m), HBLC: height of the base of the crown (insertion of the first whorl with living needles that is part of the crown as a whole) (m), R ²: coefficient of determination, REMC: mean square error.

Table 3 shows the parameter estimators of the three models and their fit statistics using normal diameter as the predictive variable. In all cases the parameters were significant at the 5 % significance level.

Table 3  - Estimated parameters and fit statistics of the models used to model the maximum crown diameter of Pinus cooperi, depending on the maximum crown diameter  

Model	Parameters		Standard error	REMC	R^2
$\left[1\right]$	${\beta }_0$	1,7574	0,4465	1,1092	0,819
	${\beta }_1$	0,1608	0,0094
$\left[2\right]$	${\beta }_0$	-1,1081	1,2764	1,0700	0,831
	${\beta }_1$	0,2889	0,0544
	${\beta }_2$	-0,00129	0,00054
$\left[3\right]$	${\beta }_1$	0,4342	0,0805	1,0906	0,824
	${\beta }_2$	0,7986	0,0472

R²: coefficient of determination, REMC: root mean square error.

The Figure 3 shows a homogeneous distribution of the model errors, indicating that there is homoscedasticity, so the assumptions for the regression analysis are fulfilled.

Fig. 3.  - Predicted values versus residuals of the three models analyzed 

On the other hand, Figure 4 shows the graphs of the predicted values against the observed values, observing that the three models provide a good prediction of the maximum crown diameter using the normal diameter as an independent variable.

Fig. 4.  - Maximum observed crown diameters versus predicted values obtained through the use of the three models using the normal diameter as the predictive variable. The solid line represents a linear model fitted to the scatter plot 

Although the fit of the three models presents very little variation in the coefficient of determination and in the precision estimator (REMC), the regression equation derived from the second-degree polynomial model provides slightly better predictions which has already been reported in other research (^{Sharma et al., 2017}; ^{Bera et al., 2021}), therefore, its use is recommended to estimate the maximum crown diameter of Pinus cooperi through the following expression Equation 6:

$dmc=-1.1081+0.2889*dn-0.00129*{dn}^2$ (13)

Where:

dmc	maximum crown diameter (m)
dn	normal diameter (m)

The inclusion of dn² in the model is justified since much of the existing literature to predict crown diameter via normal diameter (^{Bechtold, 2003}; ^{Hasenauer, 1997}; ^{Lhotka and Loewenstein, 2008}; ^{Martin et al., 2012}; ^{Chen et al., 2021}), as well as previously published maximum crown diameter equations have used dn² (^{Paine and Hann 1982}; ^{Smith et al., 1992}; ^{Sporek and Sporek, 2023}).

With the values of the estimators and multiplying by the ratio, $\frac{\pi }{4}$ we have the equation for the maximum crown area (Amc) (^{Hutch et al., 1993}; ^{Di Salvatore et al., 2021}) Equation 7:

$Amc=\frac{\pi }{4}({-1.1081+0.2889*dn-0.00129*{dn}^2)}^2$ (14)

Where:

Amc	maximum crown area (m^2 )

Amc equation for a range of trees with a diameter of 20 to 80 cm, the estimated maximum crown area for Pinus cooperi varies from 13.55 m ² (0.136 % cover) to 156.77 m ² (1.568%) cover. In turn, the minimum average number of trees necessary to cover 100 % of the surface of a hectare by diameter category within the interval of 20 to 85 cm varied from 738 (23.18 m² ha^-1 ) to 64 (36.20 m² ha^-1), respectively (Table 4). This number of trees is slightly lower than those estimated with the equation reported by ^{Quiñones and Ramírez (1998}) for the same species that is present in a region neighboring the one studied. For Pinus cooperi, the average number of trees required to cover 100 % of the surface of a hectare by diameter category within the interval of 20 to 85 cm varied from 738 (23.18 m² ha^-1) to 64 (36.20 m² ha^-1), respectively (Table 4). This number of trees is slightly lower than those estimated with the equation reported by ^{Quiñones and Ramírez (1998)} for the same species that is present in a region neighboring the one studied. To Pinus rudis Endl. in Oaxaca, similar results were obtained in the diameter category of 20 with 740 trees (^{Martínez et al., 2021}). According to this equation, between 627 and 80 trees per hectare are estimated within the range of the diameter categories studied. ^{Hernández et al . (2013}) report that in Pinus teocote Schlecht. Et Cham. from the state of Hidalgo the number of trees decreases from 580 to 55 within the range of 20 to 85 cm of normal diameter, while ^{Rodriguez et al. (2009}) mentions that in Pinus montezumae Lamb. these decrease from 557 to 168 within a range of 20 to 40 cm of normal diameter.

Table 4.  - Density attributes from the estimation of the maximum crown area of Pinus cooperi 

Dn (cm)	AMC ( ^m2 )	Arbha ^-1	ABha ^-1 (m ^2 ha ^_1 )	ACi (%)
20	13.55	738	23,18	0.136
25	22,13	452	22,18	0.221
30	32,15	311	21,99	0,321
35	43,28	231	22,23	0,433
40	55,21	181	22,76	0,552
45	67,64	148	23,51	0,676
50	80,31	125	24,45	0,803
55	92,96	108	25,56	0,930
60	105,35	95	26,84	1,054
65	117,29	85	28,29	1,173
70	128,56	78	29,94	1,286
75	139,00	72	31,78	1,390
80	148,44	67	33,86	1,484
85	156,77	64	36,20	1,568

Dn : normal diameter, AMC: maximum crown area, Arbha ^-1: trees per hectare, Abha ^-1: basal area per hectare, ACi : percentage cover

The density graph created from the CCF (Figure 5) indicates the threshold of the minimum number of trees per diameter category that is necessary to cover 100 % of the surface. From the point of view of density management for timber production purposes, the 100 % CCF line is the basis for estimating the level of competition between individuals and optimal land use. A CCF less than 100 % indicates the absence of competition and deficiency in land use, so to optimize the quality and quantity of timber production it is desirable to maintain the density of a forest above 100 % coverage.

Fig. 5.  - Relationship between the number of trees per diameter category at different levels of crown competition factor (CCF) 

CONCLUSIONS

Normal diameter is the most explanatory variable in a simple linear model and a quadratic one and best predictor for maximum crown diameter in Pinus cooperi .

The quadratic polynomial model best describes the normal diameter-maximum crown diameter relationship, so it is recommended for use in the study of growth and competition of this species in even-aged and mixed and irregular stands in the state of Durango.

Two forest productivity scenarios can be identified related to the crown competition factor; when it is 100 % or higher, timber productivity is optimal; and if it is less than 100 %, productivity is not optimal.

ACKNOWLEDGEMENTS

Our thanks to the Regional Forest Management Unit 1008 (UMAFOR 1008) of the state of Durango for the facilities provided in carrying out this research.

REFERENCIAS BIBLIOGRÁFICAS

ARNONI, E., GUIMARÃES, C.A., SCHNEIDER, P.R., HESS, A., LIESENBERG, V., y TAGLIAPIETRA, C., 2020. Modelado de índices de competencia para Araucaria angustifolia en dos sitios en el sur de Brasil. Bosque (Valdivia), [En línea] Vol. 41, no. 1 [Consulta: 01/09/2024]. ISSN 0717-9200. DOI: 10.4067/s0717-92002020000100065. Disponible en: Disponible en: http://dx.doi.org/10.4067/S0717-92002020000100065

BERA, D., CHATTERJEE, N., y BERA, S., 2021. Comparative performance of linear regression, polynomial regression and generalized additive model for canopy cover estimation in the dry deciduous forest of West Bengal. Remote Sensing Applications: Society and Environment [En línea]. Vol. 22 [Consulta: 18/08/2024]. ISSN 2352-9385. DOI: 10.1016/j.rsase.2021.100502. Disponible en: Disponible en: https://doi.org/10.1016/j.rsase.2021.100502 .

BECHTOLD, W.A., 2003. Crown-Diameter Prediction Models for 87 Species of Stand-Grown Trees in the Eastern United States. Southern Journal of Applied Forestry, [En línea]. Vol. 27, n.º 4 [Consulta: 08/09/2024]. ISSN 1938-3754. DOI: 10.1093/sjaf/27.4.269. Disponible en:Disponible en:https://doi.org/10.1093/sjaf/27.4.269 .

BIGING, G., y DOBBERTIN, M., 1995. Evaluation of competition indices in individual tree growth models. Forest Science [En línea]. Vol. 41, n.°2 [Consulta: 18/09/2024. ISSN 1938-3738. DOI: 10.1093/forestscience/41.2.360. Disponible en: Disponible en: https://doi.org/10.1093/forestscience/41.2.360

BRAGG, D. C., 2001. A local basal area adjustment for crown width prediction. Northern Journal of Applied Forestry [En Linea]. Vol. 18, n.° 1. [Consulta: 25/08/2024]. ISSN 1938-3762. DOI: 10.1093/njaf/18.1.22. Disponible en: Disponible en: https://doi.org/10.1093/njaf/18.1.22

BRAVO, A., MARCHI, M., TRAVAGLINI, D., PELLERI, F., MANETTI, M.C., CORONA, P., et al., 2020. Adoption of new silvicultural methods in Mediterranean forests: the influence of educational background and sociodemographic factors on marker decisions. Annals of Forest Science [En línea], vol. 77, n.º 2 [Consulta: el 21/09/2024]. ISSN 1297-966X. DOI: 10.1007/s13595-020-00947-z. Disponible en: Disponible en: https://doi.org/10.1007/s13595-020-00947-z .

BUENO, G.F., COSTA, E.A., FINGER, C.A.G., LIESENBERG, V., y BISPO, P.D.C., 2022. Machine learning: crown diameter predictive modeling for open-grown trees in the cerrado biome, Brazil. Forests [En línea], Vol. 13, n.° 8 [Consulta: el 12/08/2024]. ISSN 1999-4907. DOI: 10.3390/f13081295. Disponible en: Disponible en: https://doi.org/10.3390/f13081295

CABON, A., MOUILLOT, F., LEMPEREUR, M., OURCIVAL, J.M., SIMIONI, G., y LIMOUSIN, J.M., 2018. Thinning increases tree growth by delaying drought-induced growth cessation in a Mediterranean evergreen oak coppice. Forest Ecology and Management [En línea]. Vol. 409 [Consulta: 09/09/2024] ISSN 0378-1127. DOI: 10.1016/j.foreco.2017.11.030. Disponible en: Disponible en: https://doi.org/10.1016/j.foreco.2017.11.030 .

CHEN, Q., DUAN G., LIU Q., YE Q., SHARMA R.P.., CHE Y, LIU H., y FU, L., 2021. Estimating crown width in degraded forest: A two-level nonlinear mixed-effects crown width model for Dacrydium pierrei and Podocarpus imbricatus in tropical China. Forest Ecology and Management [En línea]. Vol. 497 [Consulta: 20/09/2024]. ISSN 1999-4907. DOI: 10.3390/f13081295. Disponible en:Disponible en:https://doi.org/10.3390/f13081295

CISNEROS, A.B., MOGLIA, G., y ÁLVAREZ, J.A., 2019. Morfometría de copa en Prosopis alba Griseb. Ciência Florestal [En linea]. Vol. 29, n.°2. [Consulta: 29/08/2024]. ISSN 1980-5098. DOI: 10.5902/1980509826846. Disponible en: Disponible en: https://doi.org/10.5902/1980509826846 .

COLÍN, J.G., AGUIRRE, O.A., CORRAL, J.J., VIVEROS, E., CORRAL, S., y CRECENTE, F., 2018. Influence of competition on the diametric growth of Pinus durangensis Martínez in Durango, Mexico. Revista Mexicana de Ciencias Forestales [En línea]. Vol. 9, n.°45 [Consulta: 10/07/2024]. ISSN 2448-6671. DOI: 10.29298/rmcf.v9i45.145 Disponible en: Disponible en: https://doi.org/10.29298/rmcf.v9i45.145

CONDES, S., y STERBA, H., 2005. Derivation of compatible crown width equations for some important tree species of Spain. Forest Ecology Management [En línea]. Vol. 217 n.°2 [Consulta: 28/07/2024]. ISSN 1999-4907. Pp. 203-218 Disponible en. Disponible en. https://doi.org/10.1016/j.foreco.2005.06.002

COOMBES, A., MARTIN, J., y SLATER, D., 2019. Defining the allometry of stem and crown diameter of urban trees. Urban Forestry & Urban Greening [En línea]. Vol. 44. [Consulta: 17/09/2024]. ISSN 1618-8667. DOI: 10.1016/j.ufug.2019.126421. Disponible en; Disponible en; https://doi.org/10.1016/j.ufug.2019.126421

CORRAL, J.J., ÁLVAREZ, J.G., RUÍZ, A.D., y GADOW, K., 2004. Compatible height and site index models for five pine species in El Salto, Durango (Mexico). Forest Ecology and Management [En línea]. Vol. 201. [Consulta: 08/06/2024]. ISSN 1999-4907. DOI: 10.1016/j.foreco.2004.05.060. Disponible en: Disponible en: https://doi.org/10.1016/j.foreco.2004.05.060

CROOKSTON, N.L., y STAGE, A.R., 1999. Percent canopy cover and stand structure statistics from the Forest Vegetation Simulator. US Department of Agriculture, Forest Service, Rocky Mountain Research Station: Ogden, UT, USA, 1999. 63 p. Disponible en https://www.fs.usda.gov/fmsc/ftp/fvs/docs/gtr/percancv.pdf

CURTIS, R.O., y REUKEMA, L., 1970. Crown development and site estimates in a Douglas fir plantation spacing test. Forest Science [En linea]. Vol. 16, n.°3 [12/07/2024]. ISSN 1938-3738. DOI: 10.1093/forestscience/16.3.287. DOI: 10.1093/forestscience/16.3.287. Disponible en; Disponible en; https://doi.org/10.1093/forestscience/16.3.287

DI SALVATORE, U., MARCHI, M., y CANTIANI, P., 2021. Single-tree crown shape and crown volume models for Pinus nigra JF Arnold in central Italy. Annals of Forest Science [En línea]. Vol. 78 [Consultado 03/09/2024]. ISSN 1297-966X. DOI: 10.1007/s13595-021-01099-4. Disponible en: Disponible en: https://doi.org/10.1007/s13595-021-01099-4

GIVNISH, T.J., 2020. The adaptive geometry of trees. The American Naturalist [En línea]. Vol. 195. N.°6 [Consulta: 11/09/2024]. ISSN 1537-5323. DOI: 10.1086/708498. Disponible en: Disponible en: https://doi.org/10.1086/708498 .

GONZÁLEZ-BENECKE, C.A., FERNÁNDEZ, M.P., GAYOSO, J., PINCHEIRA, M., y WIGHTMAN, M., 2022. Using tree height, crown area and stand-level parameters to estimate tree diameter, volume, and biomass of Pinus radiata, eucalyptus globulus and eucalyptus nitens. Forests [En línea]. Vol. 13, n.°12 [Consulta: 29/08/2024]. ISSN 1999-4907. .DOI: 10.3390/f13122043 Disponible en: Disponible en: https://doi.org/10.3390/f13122043

HASENAUER, H. (1997). Dimensional relationships of open-grown trees in Austria. Forest Ecology and Management [En línea]. Vol. 96, n.° 3 [Consulta: 18/08/2024]. ISSN 1999-4907. DOI: 10.1016/S0378-1127(97)00057-1. Disponible en: Disponible en: https://doi.org/10.1016/S0378-1127(97)00057-1

HASENAUER, H., MOSER, M., y ECKMU¨LLNER, O., 1994. MOSES-a computer simulation program for modelling stand response. In: Pinto da Costa, M.E., Preuhsler, T. (Eds.), Mixed Stands, Research Plots, Measurement and Results, Models. Inst. Superior de Agronomia, University Technica de Lisboa, Lisboa Codex, Portugal (Appendix 1:1).

HERNÁNDEZ, J., GARCÍA, J.J., MUÑOS, H.J., GARCÍA, X., SÁENS, T., FLORES, C., y HÉRNÁNDEZ, A., 2013. Guía de densidad para manejo de bosques naturales de Pinus teocote Schlecht. Et Cham. En Hidalgo. Revista Mexicana de Ciencias Forestales, [En línea]. Vol. 4, n.° 19 [Consulta: 19/08/2024]. ISSN 2448-6671. Disponible en: Disponible en: https://www.scielo.org.mx/scielo.php?pid=S2007-11322013000500006&script=sci_arttext

HERNÁNDEZ, J., REYES, V.J., y BELTRÁN, L., 2022. La copa como indicador fotosintético relevante en el manejo forestal de bosques templados. Revista Mexicana de Ciencias Forestales [En linea]. Vol 13, n.° 74 [Consulta: 10/09/2024]. ISSN 2448-6671. DOI: 10.29298/rmcf.v13i74.1267. Disponible en: Disponible en: https://doi.org/10.29298/rmcf.v13i74.1267

HESS, A.F., LOIOLA, T., ARRUDA DE SOUZA, I., y NASCIMENTO, B., 2016. Morfometría de la copa de Araucaria angustifolia en sitios naturales en el sur de Brasil. Bosque (Valdivia) [En línea]. Vol. 37, n.° 3 [Consulta 20/08/2024]. ISSN 0717-9200. DOI: 10.4067/S0717-92002016000300017. Disponible en; Disponible en; http://dx.doi.org/10.4067/S0717-92002016000300017

HUSCH, B., MILLER, C., y BEERS, T., 1993. Forest mensuration. Krieger Publishing Co. Malabar, Florida, USA, 410 p. Disponible en: https://books.google.es/books?hl=es&lr=&id=p0v3m8Pau-kC&oi=fnd&pg=PR11&dq=Forest+mensuration&ots=LJFOmoHGVu&sig=6fep_KKHmxk2UCe_poUIlw9-yDc#v=onepage&q=Forest%20mensuration&f=false

INSTITUTO NACIONAL DE ESTADÍSTICA GEOGRAFÍA E INFORMÁTICA (INEGI), 2015. Anuario Estadístico Durango [En línea]. [consulta: 26/08/2024]. Disponible en: Disponible en: http://www.datatur.sectur.gob.mx/ITxEF_Docs/DGO_ANUARIO_PDF15.pdf

JUCKER, T., CASPERSEN, J., CHAVE, J., ANTIN, C., BARBIER, N., BONGERS, F., et al., 2017. Allometric equations for integrating remote sensing imagery into forest monitoring programmes. Global change biology [En línea]. Vol. 23, n.°1 [Consulta: 19/08/2024]. ISSN 1354-1013. DOI: 10.1111/gcb.13388 Disponible en: Disponible en: https://doi.org/10.1111/gcb.13388

KRAJICEK, J., BRINKMAN, K., y GINGRICH, S., 1961. Crown competition-a measure of density. Forest Science [En línea]. Vol. 7 [Consulta: 08/09/2024]. ISSN 1938-3738. DOI: 10.1093/forestscience/7.1.35. Disponible en: Disponible en: https://doi.org/10.1093/forestscience/7.1.35

LHOTKA, J.M., y LOEWENSTEIN, E.F., 2008. An Examination of Species- Specific Growing Space Utilization. Canadian Journal of Forest Research [En línea]. Vol. 38, n.° 3 [Consulta: 11/08/2024]. ISSN 1208-6037. DOI: 10.1139/X07-147. Disponible en:Disponible en:https://doi.org/10.1139/x07-147 .

LI, Y., WANG, W., ZENG, W., WANG, J., y MENG, J., 2020. Development of Crown Ratio and Height to Crown Base Models for Masson Pine in Southern China. Forests [En línea]. Vol. 11, n.° 11 [Consulta 23/08/2024]. ISSN 1999-4907. DOI: 10.3390/f11111216. Disponible en: Disponible en: https://doi.org/10.3390/f11111216

MARCHI, M., SCOTTI, R., RINALDINI, G., y CANTIANI, P., 2020. Taper Function for Pinus nigra in Central Italy: Is a More Complex Computational System Required? Forests [En línea]. Vol. 11, n.º 4 [Consulta: 18/08/2024]. ISSN 1999-4907. DOI: 10.3390/f11040405. Disponible en: Disponible en: https://doi.org/10.3390/f11040405 .

MARTIN, N.A., CHAPPELKA, A., LOEWENSTEIN, E.F., KEEVER, G.J., y SOMERS, G., 2012. Predictive open-grown crown width equations for three oak species Planted in a Southern Urban Locale. Arboriculture & Urban Forestry [En línea]. Vol. 38, n.° 2 [Consulta: 29/07/2024]. ISSN 2155-0778. DOI: 10.48044/jauf.2012.010. Disponible en: Disponible en: https://doi.org/10.48044/jauf.2012.010

MARTÍNEZ, D., CABALLERO, P., FILIO, E., GARZÓN, A., ORTIZ, R., CRUZ, O.L., APARICIO, C., y SANDOVAL, C., 2021. Guía de densidad para el manejo de rodales naturales de Pinus rudis Endl. en Oaxaca. Revista Mexicana de Ciencias Forestales [En línea]. Vol. 12, n.° 64 [Consulta 17/09/2024]. ISSN 2448-6671. DOI: 10.29298/rmcf.v12i64.822. Disponible en linea: Disponible en linea: https://doi.org/10.29298/rmcf.v12i64.822

MOEUR, M., 1981. Crown width and foliage weight of northern Rocky Mountain conifers. [En linea]. USDA Forest Service Research Paper, INT-283, 14p. [Consulta: 01/08/2024]. Disponible en: Disponible en: https://books.google.es/books?hl=es&lr=&id=PBGOcgtHiaYC&oi=fnd&pg=PA1&dq=Crown+width+and+foliage+weight+of+northern+Rocky+Mountain+conifers.+USDA+Forest+Service+Research+Paper,+&ots=CnUq0F6PM3&sig=tGdl2DwEN9Txy3lQ2BxWQMUX2jQ#v=onepage&q=Crown%20width%20and%20foliage%20weight%20of%20northern%20Rocky%20Mountain%20conifers.%20USDA%20Forest%20Service%20Research%20Paper%2C&f=false

OU, Y., y QUIÑÓNEZ, G., 2023. Modeling Height-Diameter Relationship Using Artificial Neural Networks for Durango Pine (Pinus durangensis Martínez) Species in Mexico. Forests [En línea]. Vol. 14, n.° 8 [Consulta: 12/09/2024]. ISSN 1999-4907. DOI: 10.3390/f14081544. Disponible en: Disponible en: https://doi.org/10.3390/f14081544

PAINE, D.P., y HANN, D.W., 1982. Maximum crown-width equations for southwestern Oregon tree species [En línea]. Forest Research Laboratory, Oregon State University, Corvallis. Research Paper. 46, 20 p. [Consulta: 12/08/2024]. Disponible en: Disponible en: https://ir.library.oregonstate.edu/downloads/b2773w92s

PRETZSCH, H., AHMED, S., JACOBS, M., SCHMIED, G., y HILMERS, T., 2022. Linking crown structure with tree ring pattern: methodological considerations and proof of concept. Trees [En línea]. Vol. 36, n.° 4 [Consulta: 09/09/2024]. ISSN 1432-2285. DOI: 10.1007/s00468-022-02297-x. Disponible en: Disponible en: https://doi.org/10.1007/s00468-022-02297-x

PRETZSCH, H., BIBER, P., y DURSKY, J., 2002. The single tree-based stand simulator SILVA: Construction, application and evaluation. Forest Ecology Management [En linea]. Vol. 162, n.°1 [Consulta: 13/08/2024]. ISSN 1999-4907. DOI: Disponible en: Disponible en: https://doi.org/10.1016/S0378-1127(02)00047-6

QUADRI, O.A., 2019. Development and evaluation of linear and non-linear models for diameter at breast height and crown diameter of TriplochitonScleroxylon (K. Schum) Plantations in Oyo State, Nigeria. Journal of Agriculture and Veterinary Science [En línea]. Vol. 12, n.° 6 [Consulta: 12/09/2024]. ISNN 2319-2380. DOI: 10.9790/2380-1206014752. Disponible en: Disponible en: https://doi.org/10.9790/2380-1206014752

QUIÑONES, A., y RAMÍREZ, H., 1998. Evaluación de la calidad de sitio y del efecto de la densidad en bosques de Pinus cooperi del estado de Durango. Revista Chapingo Serie Ciencias Forestales y del Ambiente. Vol. 4, n.° 2 [Consulta: 10/08/2024]. ISSN 2007-4018.

QIU, S., LIANG, R., WANG, Y., LUO, M., y SUN, Y., 2022. Comparative analysis of machine learning algorithms and statistical models for predicting crown width of Larix olgensis. Earth Science Informatics [En línea] Vol. 15 n.° 4 [Consulta: 10/09/2024]. ISSN 1865-0481. DOI: Disponible en: 10.1007/s12145-022-00854-z Disponible en: 10.1007/s12145-022-00854-z https://doi.org/10.1007/s12145-022-00854-z

REINEKE, L.H., 1933. Perfecting a stand density index for even aged forest. Journal of Agricultura Research [En línea] Vol. 46, n.° [Consulta: 18/08/2024]. ISSN 0095-9758. Disponible en: Disponible en: https://cmapspublic3.ihmc.us/rid=1N4TSLMJK-JNTN9D-14QK/Perfecting%20a%20stand-density%20index%20for%20even-aged%20forests.PDF

RODRÍGUEZ, L.R., RAZO, R., DÍAZ, H., y MEZA, J., 2009. Guía de densidad para Pinus montezumae en su área de distribución natural en el Estado de Hidalgo. [Consulta: 15/09/2024]. Folleto Técnico No. 1. Universidad Autónoma del Estado de Hidalgo. Instituto de Ciencias Agropecuarias. Área Académica de Ingeniería Forestal. Tulancingo de Bravo, Hgo. México. 31 p. Disponible en: Disponible en: https://www.uaeh.edu.mx/investigacion/icap/LI_IntGenAmb/Rodri_Laguna/1.pdf

SANTIAGO-GARCÍA, W., DE LOS SANTOS, P.H.M., ÁNGELES G., VALDEZ J.R., DEL VALLE, D.H., y CORRAL, J.J. 2013. Auto-aclareo y guías de densidad para Pinus patula mediante el enfoque de regresión de frontera estocástica. Agrociencia [En linea]. Vol. 47, n.° 1 [Consulta: 29/08/2024]. ISNN 2521-9766. Disponible en: Disponible en: https://www.scielo.org.mx/scielo.php?pid=S1405-31952013000100007&script=sci_arttext

SECRETARÍA DEL MEDIO AMBIENTE Y RECURSOS NATURALES (SEMARNAT), 2019. Producción forestal maderable y no maderable. [En línea]. [consulta: 16/08/2024]. Disponible en: Disponible en: https://apps1.semarnat.gob.mx:445/dgeia/indicadores17/conjuntob/indicador/07_f orestales/7_2.html

SHARMA, R.P., BÍLEK, L., VACEK, Z., y VACEK, S., 2017. Modelling crown width-diameter relationship for Scots pine in the central Europe. Trees [En línea]. Vol. 31 [Consulta: 28/08/2024]. ISSN 1432-2285. DOI: 10.1007/s00468-017-1593-8, Disponible en; Disponible en; https://doi.org/10.1007/s00468-017-1593-8

SPOREK, M., y SPOREK, K., 2023. Allometric Model of Crown Length for Pinus sylvestris L. Stands in South-Western Poland. Forests [ En linea]. Vol. 14, n.° 9 [Consulta: 16/08/2024] ISSN 1999-4907. DOI: 10.3390/f14091779. Disponible en: Disponible en: https://doi.org/10.3390/f14091779

SMITH, W.R., FARRAR, J., y MURPHY, P.A., 1992. Crown and basal area relationships of open-grown southern pines for modelling competition and growth. Canadian Journal of Forest Research [En línea]. Vol. 22, n.° 3 [Consulta: 10/09/2024]. ISSN 1208-6037, DOI: 10.1139/x92-044. Disponible en: Disponible en: https://doi.org/10.1139/x92-044

SOTO, J.A., LÓPEZ, C.A., CORRAL, J.J., WEHENKEL, C., ÁLVAREZ, J.G., y CRECENTE, F. 2016. Desarrollo de un modelo de perfil de copa para Pinus cooperi Blanco en la UMAFOR 1008, Durango, México. Revista Chapingo Serie Ciencias Forestales y del Ambiente [En linea]. Vol. 22, n.° 2 [Consulta 06/09/2024]. ISSN 2007-4018. DOI: 10.5154/r.rchscfa.2015.09.040. Disponible en: Disponible en: https://doi.org/10.5154/r.rchscfa.2015.09.040

TAMARIT, U.J.C., QUIÑONEZ, B.G., y HERNÁNDEZ, R.J., 2020. Aspectos metodológicos para generar diagramas de manejo de la densidad de rodales con base en el índice de Reineke. Revista Mexicana de Ciencias Forestales [En línea]. Vol 11, n.° 61 [Consulta 10/09/2024].ISSN 2448-6671. DOI: 10.29298/rmcf.v11i61.728. Disponible en: Disponible en: https://doi.org/10.29298/rmcf.v11i61.728

VOSPERNIK, S., y STERBA, H., 2015. Do competition-density rule and self-thinning rule agree?. Annals of Forest Science [En línea]. Vol. 72 [ Consulta: 10/09/2024]. ISSN 1297-966X. DOI: 10.1007/s13595-014-0433-x Disponible en; Disponible en; https://doi.org/10.1007/s13595-014-0433-x

YANG, S.I., y BRANDEIS, T.J., 2022. Estimating maximum stand density for mixed-hardwood forests among various phy siographic zones in the eastern US. Forest Ecology and Management [En línea]. Vol. 521[Consulta: 18/09/2024]. ISSN 1999-4907. DOI: Disponible en: Disponible en: https://doi.org/10.1016/j.foreco.2022.120420

YANG, Y., y HUANG, S. 2017. Allometric modelling of crown width for white spruce by fixed-and mixed-effects models. The Forestry Chronicle [En línea]. Vol. 93, n.° 2 [ 06/09/2024]. ISSN 1499-9315. DOI: 10.5558/tfc2017-020. Disponible en: Disponible en: https://doi.org/10.5558/tfc2017-020

Received: July 25, 2024; Accepted: August 04, 2024

^*Autor para la correspondencia: [e-mail]

Los autores declaran no tener conflictos de intereses.

Los autores han participado en la redacción del trabajo y análisis de los documentos.