Fundamental concepts

The relevant concepts within the context of stand density management and the construction of a DMD with Reineke’s model (^{Reineke, 1933}) are described below.

Stand density. In a forest site, stand density refers to the degree of space occupation that the trees have at a given time (^{Daniel et al., 1979}; ^{Clutter et al., 1983}; ^{Smith et al., 1997}); it is a variable that through proper management helps to predict and improve some stand attributes. Density is manipulated to favorably influence the establishment and development of the species of interest; to improve the quality of wood, as well as the rate of diameter growth and the production of wood volume. Density can be expressed and assessed in absolute terms per unit area (number of living trees, basimetric area or volume), in which case the determination is direct and without reference to any other stand. It can also be expressed in relative terms by index values that determine the level of density and competence of a stand, such as Reineke’s stand density index (1933). In this form, density is expressed based on a previously selected standard density.

Reineke’s density-size function (1933). According to ^{Smith et al. (1997)}, ^{Pretzsch (2009)} and ^{Burkhart and Tomé (2012)}, it is a potential type relationship expressed as the number of trees per hectare of a stand (N), and the size of individuals, given by the quadratic mean diameter (QMD) (1).

Where

α = Intercept parameter

β = Parameter of the slope

For ease of adjustment and reduction of heteroscedasticity, this function is used in its linear form by means of logarithms (ln) (2).

^{Reineke (1933)} established that the theoretical value of the slope for any species is -1.605.

Quadratic mean diameter (QMD). This is the diameter of the tree with the average basal area of the stand (AB, m² ha^-1), calculated with the Expression (3) (^{Daniel et al., 1979}; ^{Clutter et al., 1983}; ^{Weiskittel et al., 2011}).

Where:

π = The constant 3.1416

40 000 = It is used to express AB in m² when the normal diameter is expressed in cm

N = Number of trees ha^-1

Reference quadratic mean diameter (QMDR). It is the diameter that is predefined to estimate the SDI; it can be the value suggested by ^{Reineke (1933)}, equivalent to 25 cm, or the value of the QMD observed in plots or sampling sites; even, any other value, since it is a self-reference model (^{Clutter et al., 1983}; ^{Pretzsch, 2009}; ^{Burkhart and Tomé, 2012}).

Reineke’s Stand Density Index (SDI). This is the number of trees per hectare (or acre) that a stand can have for the predefined QMDR (^{Pretzsch, 2009}; ^{Weiskittel et al., 2011}; ^{Burkhart and Tomé, 2012}); it is calculated with the Expression (4).

An alternate and equivalent way of calculating it is through the Expression (5)

When the value of ( is unknown, it is possible to use the theoretical value 1605. The relative density is expressed in relation to the preset QMDR.

Maximum Stand Density Index (SDI _max ). It is maximum density in terms of the number of trees per unit area that can exist in a population under self-thinning for a preset QMDR (example QMDR=25) and specific taxon (^{Pretzsch, 2009}; ^{Weiskittel et al., 2011}; ^{Burkhart and Tomé, 2012}). The SDI _max is estimated by the Expression (6).

Where:

e = Exponential function

ln = Natural logarithm

Upper limit line for self-sunlighting (for maximum competition or simply self-sunlighting line) This is the line given by the maximum number of trees (N _max ) for each QMD or diameter category, representing 100 % or the SDI _max (^{Pretzsch, 2009}; ^{Weiskittel et al., 2011}; ^{Burkhart and Tomé, 2012}), is determined by the Expression (8).

Information sources for variables N and QMD

The mensuration information used in most of the classical works on density to delimit the maximum density line and to build a DMD should preferably come from permanent experimental plots, located in stands with extreme density and competition; a network of such plots allows monitoring, in time, the behavior of both crown closure and the occurrence of mortality periods (^{Weller, 1987}; ^{Zeide, 1987}); however, due to long-term planning, the costs involved and the time required for the establishment and monitoring of the network, it is difficult to have sufficient mensuration data in the form of complete time series in the short and medium term (^{Pretzsch, 2006}; ^{Condés et al., 2017}). Some recent density studies utilize novel adjustment techniques such as boundary functions, using all available information from sampling sites or temporary plots (^{Lopes et al., 2016}; ^{Quiñonez et al., 2018}; ^{Salas and Weiskittel, 2018}). However, it should always be ensured that the data set contains observations on maximum density.

Under the classical approach of working density with Reineke's model, another way of obtaining the information to be processed is by means of a directed sampling oriented to temporary plots, of circular, rectangular or with any other shape. In this case, selective stratified sampling is required; the useful strata are those corresponding to the stands with the highest population or density; which leads to the selection of stands with closed coverage and high densities (^{Valencia, 1994}).

The plots will be located in pure and contemporaneous stands, with maximum density (normal or complete), without exhibiting physical damage, with healthy trees, and free of phytosanitary problems, preferably not intervened in the last 10 years; all the diameter categories, age classes and qualities of the season in which the species of interest is developed must be included. The plot area can vary according to the size of the trees; if they are adults, an area of 1 000 m² is recommended; if it holds a stage of growth and development (seedling, sapling, thicket, or pole), then the amount of trees is high and the area can be reduced, for example from 1 000 to 500 or even less m² (^{Valencia, 1994}; ^{Gezán et al., 2007}; ^{Navarro et al., 2011}).

The most practical, economical, easy and fast alternative is to use the mensuration data information that come from classic sampling sites used in operational timber inventories, which are necessarily made for the purpose of developing and implementing management programs in a wide range of growth conditions (^{Weiskittel et al., 2009}). In this case, it is important to verify that the sample is representative of the study region and for the taxon of interest.

^{Navarro et al. (2011)} state that appropriately selected high density sites are useful to determine the self-selected line, although a mixture of information from permanent and temporary sample units can be used; ^{van Laar and Akça (2007}, ^{Vospernik and Sterba (2015)} and ^{Salas and Weiskittel (2018)} ratify that, in the lack of remeasurement of permanent plots, mensuration information from temporary plots or inventory sites is adequate, after a selection treatment to delimit the self-thinning line.

These sites or plots replace the temporal succession of information from the permanent plots with a series of spatial-type mensuration information, distributed at different stages of development. Since they are only surveyed on one occasion, there is a collection of data on different stages of forest development with point measurements; therefore, a set of these sites successfully replaces a permanent plot with measurements staggered over time. However, they would have the disadvantage that it would not be possible to determine dynamic growth patterns, or mortality -essential information under some methodologies to delimit certain growth areas for the density guidelines.

High Density Site Selection

Within the context of traditional stand density analysis, not all sampling sites for operational inventory are necessarily established in extremely competitive, high-density stands. Only those that meet this condition are useful for conducting density studies, as well as for adjusting Reineke’s function, defining the upper limit line of self-clearance, and constructing DMDs. Therefore, only the sites that meet this requirement and condition should be selected. For this purpose, we recommend using the method suggested by ^{Solomon and Zhang (2002)}, known as the Maximum Stand Density Index Method (SDI _max ), the steps of which are detailed below.

1. The density of each site expressed as the number of trees per site (N) is scaled to the hectare level.

2. For each site, the quadratic mean diameter (QMD) is calculated, using the Expression (4), at the hectare level.

3. For each site, Reineke’s SDI is calculated with the Expression (5), QMDR = 25 cm and β = 1.605.

4. Of all the sites, the one with the highest SDI value (SDI _max ) is located.

5. The relative density (RD) of each site is calculated by dividing its respective SDI value by the value corresponding to the site with the highest SDI:

6. Those sites whose RD values are equal to or greater than 0.60 are selected; this ensures that they meet the condition of having high density and competition levels.

Adjusting Reineke’s density-size function

The value of the slope is different from the originally proposed theoretical value of -1.605 (^{Pretzsch, 2006}; ^{VanderSchaaf and Burkhart, 2007}; ^{Comeau et al., 2010}); therefore, it must be estimated for specific regions and taxa. In this regard, different statistical methods have been developed to adjust Reineke's function and obtain the parameter estimators (^{Zhang et al., 2005}; ^{Burkhart and Tomé, 2012}; ^{Zhang et al., 2015}; ^{Salas and Weiskittel, 2018}). The Linear Ordinary Least Square (OLS) and Stochastic Border Regression (SBR) techniques are the two main techniques utilized (^{Santiago et al., 2013}).

Under the classical modeling approach, the OLS method can be successfully used to adjust the function, is robust and recognized as the best for estimating the average condition of the random variable of interest (^{Zhang et al., 2005}; ^{Comeau et al., 2010}), since it minimizes the sum of squares of the distances between the observed and predicted values; in addition, the potential problem of heteroscedasticity is minimized (^{Gezán et al., 2007}).

The average line generated with OLS passes through the middle of the observed data cloud; however, the interest is to determine the upper limit; therefore, it cannot be considered per se as a maximum density biological line derived from the density-size relationship (^{Zhang et al., 2013}); therefore, it is necessary to use some criteria that properly delimit the maximum density line at the upper boundary.

Methodological criteria for shifting the average line and charting the self-thinning line

The maximum density line (upper limit of the self-thinning) is defined at the upper border of the observed data cloud, as this is where the extreme density and competition are found, and therefore, the highest mortality of individuals. For this aim, there are different methodological criteria based on keeping the value of the slope parameter (β) fixed and increasing the value of the intercept parameter (α) of Reineke’s function, which allows a maximum α (α_max) (^{Burkhart and Tomé, 2012}; ^{Santiago et al., 2013}). Thus, a new linear function is obtained that generates a straight line parallel to the average line, which, when displaced towards the upper border, will correspond to the maximum density line.

The main criteria with a statistical basis for increasing the value of the intercept parameter and generating the self-thinning line are described below. The application of the criteria will be exemplified using the information from the adjustment of Reineke’s function (Table 1) in the Expression (1) applied to 90 sampling sites of Pinus montezumae Lamb. of the Forest Management Unit (Umafor) 2103 Teziutlán Region, in the state of Puebla, Mexico.

Table 2 shows the method of calculation for each criterion, as well as the final values of αmax according to each criterion. It has been proven that the methodological criteria referred for correcting the value of the intercept are consistent for generating the absolute biological line of the density-size relationship (^{Solomon and Zhang, 2002}; ^{Comeau et al., 2010}).

Optimal stand growth area in volume

Based on evidence from various studies on tree mortality and competition (^{Drew and Flewelling, 1979}; ^{Long, 1985}; ^{Becerra, 1986}; ^{Dean and Baldwin, 1993}; ^{Jack and Long, 1996}; ^{Powell, 1999}; ^{Vargas, 1999}; ^{Gezán et al., 2007}; ^{Shaw and Long, 2007}; ^{Navarro et al., 2011}; ^{Weiskittel et al., 2011}; ^{Long and Shaw, 2012}; ^{Müller et al., 2013}; ^{Hernández et al., 2013}), it is assumed that the theoretical lines delimiting Langsaeter's growth areas can be determined in terms of relative density (percentages of the self-thinning line). This is because it is accepted that a certain interval of the SDI _max corresponds to a particular stage of stand development (^{Kumar et al., 1995}; ^{Pretzsch, 2009}) and each of these is equivalent to the growth areas of a DMD; which shows the importance of establishing correct relative density intervals to build a DMD. The lines that make up a DMD and their determination are presented below.

Line A. It is the self-thinning line and corresponds to 100 % or SDI _max . It determines the maximum density, and thus, the competition and maximum occupation of the site. The stands on this line are in a state of over-density; it is only desirable to maintain stands in this condition, prior to final stand harvest, to maximize total net timber production.

Line B. It corresponds to the lower limit of the area in which the self-thinning occurs and determines the beginning of mortality by competition since the site is fully occupied. This phenomenon starts between 50 and 60 % of the maximum density. Lines A and B form the growth area 4, called self-thinning or imminent mortality, which begins to be critical from a value of 75 %. In order to facilitate the generation of self-thinning scenarios and to have a wide range of density management, the lower limit of this area may be set at 65 %.

Line C. It determines the lower limit of occupation of the site and normally corresponds to the moment when the canopy closure begins to occur, partial competition begins, and, therefore, immediate mortality of the individuals does not occur. This line is obtained when the density varies from 25 to 35 %, with respect to the self-thinning line; the value of 35 % is commonly utilized for building a DMD. Between lines B and C lies the growth area No. 3, which corresponds to the maximum growth in volume per hectare of the stand, the occupation of the site is considered adequate, and the maximum and constant positive growth rate. Theoretically, the optimal range for this area is between 35 and 65 %.

The percentages are likely to change and be established based on the knowledge of the forester regarding the development and growth of the species of interest; so that full site occupancy and absence of competing mortality is achieved. ^{Long (1985)} suggests that to properly delimit the upper border of the growing area, the criterion of establishing a minimum acceptable level of individual tree vigor must be used; this should be related to the ability to respond quickly to thinning. A 40 % ratio of the live crown to the total height ratio of the tree may be adequate; furthermore, it corresponds to 50 % of the SDI _max . In order to apply this criterion, it must be taken into account that the live crown diminishes as the lower branches die, due to the increased relative density and the presence of natural pruning and crown closure.

^{Penner et al. (2006)} suggest that growth area 3 corresponds to the condition of maximum periodic annual increment (PAI) of stand volume. The appropriate level of density can be determined when stability is observed in the PAI at normal diameter; thus, for a specific diameter category, the PAI will reach a maximum and tend to stabilize; because, even if space increases, diametric growth will not increase. Otherwise, when the PAI reaches a minimum and tends to remain stable, it will indicate that the limit of survival and growth has been reached; therefore, if the density augments, the increase in diameter will not be reduced further. The fundamental principle for this area is that the silvicultural regimes applied should not allow the site to be wasted by excessive thinning and that growth should not be delayed by excessive density.

Line D. This is determined in order to know the minimum level of density that a stand must have so as to achieve minimum soil coverage; thus, the stand has the capacity to develop and reach full density at later stages. Generally, it is established at 25 % of the SDI _max . Lines C and D form the growth area 2 which corresponds to the transition area and varies from 25 to 35 %.

Growth area 1 is the underutilization of the site. The trees grow freely, without competition, and in isolation and independently of each other; since the space is large for their size, it is delimited between zero and 25 % in relation to the SDI _max ; the growth rate in volume in this area is constant and proportional to density.

The four growth areas referred to above form bands of relative densities and give rise to a DMD. In order to promote optimal growth of each stand for timber production purposes, they should be maintained in area 3 by means of thinning density management (^{Drew and Flewelling, 1979}; ^{Jack and Long, 1996}; ^{Long and Shaw, 2012}).

Figure 1 shows the DMD generated with the adjustment of Reineke’s function (Table 1) and the intercept corrected with criterion B. The referred growth lines that delimit the growth areas are shown. The maximum density line indicates the maximum number of trees for different QMDs or diameter category centers at an interval that depends on the nature of maximum growth in diameter of the species of interest.

Densidad (árboles ha^-1) = Density (trees ha^-1); Diámetro cuadrático promedio = Quadratic mean diameter; IDR = SDI; IDR _máx = SDI _max .

Figure 1 DMD with the lines that delimit the growth areas. 

Final considerations

The support provided by DMDs makes it easier to make decisions to define thinning regimes, making them an important quantitative forest management tool. What the DMD suggests should be considered only as a reference and not as a strict rule. The decision to thin, as well as the type, intensity and periods between consecutive thinnings is the responsibility of the foresters, whose decision is based on the particular situation of each stand and on the productive objective of the species.

Proper density management as a function of the application of intermediate cuttings has a direct effect on the attributes of the individuals that make up the stand, such as average diameter, taper, average crown length, branch size, vigor, health, and, primarily, volume. These attributes directly influence the quality and quantity of wood and therefore its commercial value. From the above, it is evident that it is important that the technician in charge of forest management knows how to build and dispose of DMDs that are based on Reineke’s function, when the classical modeling approach is used.

DMDs built with Reineke's model are more accurate, since they use the quadratic diameter, calculated based on the normal diameter, and this is measured directly on the trees, unlike other indices that are considered less stable, such as Yoda's, which uses the average volume estimated indirectly through volume models, or the relative spacing index that uses the total height, which is obtained indirectly, or estimated with some allometric model. In both cases, a larger margin of error is incurred, and therefore its accuracy and reliability may be lower. The initially generated DMDs are basic, since only two stand attributes are represented in a two-dimensional plane; however, they are practical since they facilitate reading and interpretation to diagnose the density and competence status of the stands. The subsequent incorporation of isolines of other stand variables, such as dominant height or site index, basal area, volume, or other, into the DMD will allow for the development of more complex, yet more comprehensive and complete, density diagrams.