Study area

The agrarian community of San Miguel El Grande is located in western Oaxaca, at an altitude ranging from 2 200 to 3 330 m, between 16°58'29" and 17°10'27" N, and 97°33'23" and 97°40'21" W. The study was conducted in the natural distribution area of P. rudis, in an area of approximately 1 242 ha (Figure 1). The predominant types of climates in the community are Cb’(w₂) and C(c₂), with summer rains (^{García, 1998}). The soil types are regosols, lithosols, and vertisols (^{INEGI, 2014}). The vegetation is represented by pine and pine-oak forests (^{INEGI, 2016}), where the main species of conifers are Pinus oaxacana Mirov, P. pseudostrobus Lindl., P. douglasiana Martínez, P. rudis, P. oocarpa Schiede and P. leiophylla Schltdl. et Cham. and Abies hickelii Flous & Gaussen; taxa of the Quercus genera and Arbutus grow in lesser proportions (^{Martínez, 2009}).

Figure 1 Geographic location of the study area. 

Estimation of Reineke’s Stand Density Index

For the estimation of this index, 81 sites of different dimensions were selectively located in the following areas, using the sampling method proposed by ^{Prodan (1968)}, which considers the measurement of six trees and defines the sampling unit as a circular area whose radius extends from the center of the site to the center of the sixth tree (Figure 2). The site was selected considering areas of forest between scrubland and woodland (^{Müller et al., 2013}; ^{Aguilar, 2018}), stands of which more than 90 % consisted of trees of the same species, homogeneous areas of full occupancy, and sites with healthy trees, free of physical damage (^{Lee and Choi, 2019}).

Figure 2 Six-tree sampling site. 

The normal diameter (D) of all individuals at the sampling site was measured with a diameter tape (Forestry Suppliers, model 283d/160 cm), at a height of 1.3 m; this variable was used to estimate the basal area of the individuals present within the sampling site.

Where:

g = Individual basimetric area (m²)

D = Normal diameter (cm)

π/4 = Constant (0.7854)

Two measures of absolute density ―basimetric area per hectare (G) and number of trees per hectare (N)― were estimated using the equations described by ^{Ramos et al. (2017)}.

Where:

N = Number of individuals per hectare

N _i = Number of individuals per site

G = Basimetric area per hectare (m²)

G _i = Basimetric area per site (m²)

Sh = Area of one hectare (m²)

Se = Area evaluated at sampling site (m²)

Once these two indicators (G and N) were estimated, the quadratic diameter (Qd) of each sample plot was calculated, with the following expression (^{Santiago et al., 2013}):

Where:

Qd = Quadratic diameter (cm)

Ba = Basimetric area per hectare (m²)

Na = Number of trees per hectare

π = Constant (3.1416)

The Qd and Na values per site were used to fit Reineke’s model (5), which defined the maximum average density line of the guide (line A). For stand density comparison purposes, a Qd of 25 cm or 10 inches is considered in this index (^{Reineke, 1933}; ^{Santiago et al., 2013}; ^{Hernández et al., 2013}).

Where:

Na = Number of trees per hectare

Qd = Quadratic diameter (cm)

β ₀ = Intercept to the ordinate axis

β ₁ = Slope

Ba was the product of the individual basimetric area at a given diameter, times the number of trees estimated with Equation 5 (^{Rodríguez et al., 2009}):

Where:

Ba = Basimetric area per hectare (m²)

D = Normal diameter (cm)

Na = Number of trees per hectare

π/4 = Constant (0.7854)

Estimation of the crown competition factor

This measure of density was estimated through a targeted sampling of 54 trees that grew free of competition, healthy and without physical-mechanical defects. The variables evaluated for each individual were diameter (D) and crown diameter (Cd), which were measured with a Pretul model Pro-30me longimeter, in a north-south and east-west direction, in order to obtain the average per individual (^{Rodríguez et al., 2009}).

The 54 pairs of data (D-Cd) were used to estimate the Cd as a function of D (7), after which the crown areas (Ca) (8) corresponding to trees free of competition at a given diameter were calculated (^{Rodríguez et al., 2009}; ^{Hernández et al., 2013}).

Where:

Cd = Crown diameter (m)

D = Normal diameter (cm)

β ₀ = Intercept to the ordinate axis

β ₁ = Slope

Ca = Crown area (m²)

π/4 = Constant (0.7854)

The maximum crown area (Mca) of an individual, expressed as a percentage of the surface unit (ha) was determined with Equation 9 (^{Rodríguez et al., 2009}; ^{Hernández et al., 2013}):

Where:

Mca = Maximum crown area

Ca = Crown area (m²)

Finally, the number of trees and basimetric area per hectare were estimated with Equations 10 and 11, respectively; based on these values, the sufficient density line of the guide (line B) was calculated (^{Rodríguez et al., 2009}; ^{Hernández et al., 2013}):

Where:

Na = Number of trees per hectare

Mca = Maximum crown area

Ba = Basal area per hectare (m²)

D = Normal diameter (cm)

π/4 = Constant (0.7854)

Model fit

The adjustment of Equations 5 and 7 was performed by the Ordinary Least Squares (OLS) method in STATISTICA 10 software (^{StatSoft Inc., 2011}), through nonlinear and linear regression, respectively. The quality of fit of the models was evaluated by the value of the coefficient of determination (r ² ), the root mean squared error (RMSE), as well as the value of their likelihood (^{Vargas, 1999}; ^{Hernández et al., 2013}).

Density guide construction

In order to construct the density guide, line A or 100 % (average maximum density) was determined with the Na ha^-1 and Ba ha^-1 estimated from the Reineke model; based on these values, different density levels (30 to 110 %) were calculated at intervals of 10 %. The B line (sufficient density) was determined by the Na ha^-1 and Ba ha^-1 resulting from the CCF (^{Vargas, 1999}).

Reineke’s density index

The Qd explained 94 % (r² = 0.94) of the variation in the number of trees per hectare, with a mean error (RMSE) of 831 and significant parameters at 95 % confidence (α<0.0001). The resulting equation was as follows:

The graphical representation of the observed values (Qd and Na ha^-1) showed an inverted "J" dispersion, which is shown in Figure 3 with the fitting curve of the Reineke model (Figure 3).

Diámetro cuadrático = Quadratic diameter; Número de árboles por hectárea = Number of trees per hectare; Valores observados = Observed values; Modelo de Reineke = Reineke's model.

Figure 3 Relationship between the quadratic diameter and the number of trees per hectare. 

The values of the average maximum densities, estimated from Reineke's model, were the basis for constructing the rest of the isolines of the density guide (Table 1).

Based on the reference Qd value (25 cm), an SDI of 1 006 was estimated. In this regard, ^{Rodríguez et al. (2009)} obtained a value of 1 663 for P. montezumae Lamb; while ^{Hernández et al. (2013)} recorded a value of 775 for Pinus teocote Schiede ex Schltdl. & Cham. Both studies were conducted in the state of Hidalgo, Mexico. Approximate SDI values can be calculated only in pure stands with full density consisting of trees of the same species and with the same average stand diameter (^{Reineke, 1933}).

Several values of the slope (β ₁ ) have been estimated for the Reineke´s model, most varying between -1.0151 (^{Rodríguez et al., 2009}) and -2.18937 (^{Tamarit et al., 2018}). Likewise, studies have been developed to determine if the value of β ₁ postulated by ^{Reineke (1993)} is statistically equal to -1.605; one of them is by ^{Pretzsch and Biber (2005)}, who adjusted the equation for forests of Fagus sylvatica L., Picea abies (L.) Karst., Pinus sylvestris L. and Quercus petraea (Mattuschka) Liebl. in Germany; for these taxa, P. sylvestris L. excepted, the slope value was significantly different.

Guezan et al. (2007) developed density diagrams for Nothofagus obliqua (Mirb.) Oerst, N. alpina (Poepp. & Endl.) Oerst., N. dombeyi (Mirb.) Oerst. in Chile, and concluded that the value of the parameter β ₁ is species-specific.

^{Santiago et al. (2013)} constructed density guides for Pinus patula in the state of Hidalgo, Mexico. The estimated slopes were -1.565±0.208 for Reineke’s model and 1.199±0.048 for Yoda’s model. In the first case, it includes the value postulated by ^{Reineke (1933)}. Likewise, ^{Quiñonez et al., (2017)}, in their density plot work in mixed forests of Durango, Mexico, obtained an interval from -1.541 to -1.778, which includes the value of -1.605.

^{Santiago et al. (2013)}, ^{Quiñonez et al. (2017)} and ^{Tamarit et al. (2018)} compared adjustment methods for defining the self-thinning line and used Ordinary Least Squares (OLS) and stochastic border regression (SBR), and they agree that SBR efficiently estimates the upper boundary of self-thinning. According to ^{VanderSchaaf and Burkhart (2007)} and ^{Comeau et al. (2010)}, the OLS method is also appropriate for characterizing the maximum density line.

In general terms, the SDI has been widely used in the construction of density diagrams or guides (^{Vargas, 1999}; ^{Gezan et al., 2007}; ^{Navarro et al., 2011}; ^{Santiago et al., 2013}; ^{Hernández et al., 2013}; ^{Quiñonez et al., 2017}; ^{Tamarit et al., 2018}; ^{Tamarit et al., 2020}). These diagrams offer greater precision compared to those constructed with the Yoda index and the Relative Space index, since Reineke’s model uses the Qd calculated from the normal diameter, which is a direct measurement variable; the other two use the volume and total height, respectively, which are generally estimated using some mathematical model (^{Tamarit et al., 2020}).

Crown competition factor

Diameter (D) accounted for 83 % (r²=0.83) of the variation in Cd, with a mean error (RMSE) of 1.05 and significant parameters at 95 % confidence (α<0.0001). The resulting equation was defined as:

Table 2 shows the results obtained from Equations 7-11, whose objective was to estimate the Na ha^-1 and Ba ha^-1 in order to define the sufficient density line of the guide.

The trees in any stand below the CCF (line B) have sufficient resources to develop their full growth potential, since they do not compete with other individuals (^{Krajicek et al., 1961}; ^{Álvarez et al., 2004}). However, if the density is kept below the CCF until the trees reach their maturity, these will be of poor quality if pruning is not applied (^{Krajicek et al., 1961}). Within this context, the individuals are resistant to mechanical forces (wind and snow), but with disadvantages in sawing performance, since they have a low slenderness value and a high crown percentage (^{Arias, 2005}).

When the stand is above line B, crown closure occurs. At this time, competition between individuals begins; however, there is no immediate mortality (^{Gezan et al., 2007}). ^{Philbrook et al. (1973)} point out that the effect of competition starts when the stock is halfway between A and B, i.e., for this study, above 70 % of the SDI.

^{Santiago et al. (2013)} estimated that natural mortality in Pinus patula occurs from 55 % of the SDI up. According to ^{Quiñonez et al. (2017)}, in stands with a mixture of species, the effect of competition occurs when the value of the SDI is 70 %.

Density guide construction

Line A was determined based upon the average maximum densities estimated with Reineke’s model; stands above this line are considered overpopulated areas, which require immediate cutting to improve the quality of the residual tree stock (^{Hernández et al., 2013}; ^{Santiago et al., 2013}; ^{Quiñonez et al., 2017}; ^{Tamarit et al., 2018}).

Line B is represented by the CCF; any stand below this line is considered underpopulated or with poor density (^{Vargas, 1999}; ^{Rodríguez et al., 2009}; ^{Hernández et al., 2013}). The density of the stand of the present study must be increased so that decisions can be made regarding its management, or if necessary, another intermediate treatment ―such as fertilization, if the main objective is timber production― must be applied in order to achieve site improvement (^{Daniel et al., 1982}).

In order to exemplify the application of the guide, three sampling sites of 100 m² (10 × 10 m) were erected in an area of natural regeneration of Pinus rudis. The normal diameter of each individual was measured, and the number of trees per site was counted; based on these variables, the Ba ha^-1 and Na ha^-1 were calculated. The average values calculated were 38.27 m² of Ba ha^-1 and 2 045 m² of Na ha^-1. By placing these values in the guide (1), an SDI of 96 % and an approximate Qd of 15.5 cm were graphically estimated.

If the stand is taken from condition 1 to condition 2, its density level will rise to 75 %, and 8.26 m² of Ba ha^-1 and approximately 545 Na ha^-1 will be removed. The purpose is to maintain the stand between lines A and B in a desired condition according to management objectives (Figure 4).

Diámetro cuadrático = Quadratic diameter; Área basal = Basimetric área; Número de árboles por hectárea = Number of trees per hectare; Índice de densidad de Reineke = Reineke's density index; Deficiente = Deficient; Suficiente = Sufficient; Excesiva = Excessive.

Figure 4 Density guide for Pinus rudis Endl.