# Individual-Tree Competition Indices and Improved Compatibility with Stand-Level Estimates of Stem Density and Long-Term Production

^{1}

^{2}

^{*}

## Abstract

**:**

_{V}) that is partially derived from the Spurr (1962) competition index. Point sampling is a fast, easy, and inexpensive methodology for selecting the number of competing trees, and the new BAF

_{V}methodology proved to be very efficient for estimating growth. With the selection of an appropriate basal area factor, it is possible to reduce the number of subject trees and competition trees in the sample, and eliminate the need for edge correction methods. Although the index value assigned to the subject tree using the BAF

_{V}is biased, an appropriate correction is presented and discussed. The average competition index obtained from using the corrected value for the subject tree and the BAF

_{V}for the competing trees equals the stand level estimate of basal area.

## 1. Introduction

^{2}/4, where d is diameter at breast height (cm) and R is the distance to the center of the tree (m) from the point (plot) center. While the Spurr [7] angle summation technique has found some popularity as an individual tree competition index for growth and yield models [9], its original formulation cannot be used to duplicate a random sample of stand density in terms of basal area. Rouvinen and Kuuluvainen [10] derived an angle summation competition index that is based on the value of arctan (d/R). The advantage of the index by [10] is that the measurement equals π/2 as R → 0, whereas BLF → ∞ as R → 0.

^{2}/4. Experience in southeastern United States, however, reveals that the Haglöf DME 201 ultrasonic distance measuring instrument is ideally suited for quickly measuring distances to the nearest 0.01 m without walking from the subject tree to the competing “in” tree. In forest conditions with dense understory vegetation consisting of shrubs and arborescents, the distance of many candidate trees that are further from the point center must be verified, since it is difficult to visually verify if the neighboring tree is included “in” the sample. The added time to measure the distance of close-to-point center trees is relatively small.

_{2}≠ [V

_{1}+ ΔV], where ∆ represents volume change and V

_{i}is stand volume at time i [16]). Based on this evidence, Green [17] noted that it is difficult to detect much enthusiasm for the analysis of remeasurements using the angle count methodology.

_{AC}) created by Bitterlich [8], and the newer variable basal area factor (BAF

_{V}) [18]. The primary research objectives of this analysis were to: (1) demonstrate the application of a new estimator for quantifying long-term production; (2) develop a new tree-level competition index based on point sampling; and (3) evaluate both estimators in two contrasting plantations and examine behavior of bias, precision, and additivity.

## 2. Materials and Methods

#### 2.1. Plantation Locations and Tree Attributes

_{V}competition index. The first location was a 40 ha forest stand measured with a sampling intensity of one point (plot) per ha. The inclusion of “in” trees was determined with a BAF

_{AC}= 4 using the same point location at ages 12 and 16. The population consists of 500 loblolly pine plantation trees in Santa Catarina, Brazil generated from the growth and yield model described by McTague [19]. The modeled population is derived from a thinned 12-year old stand, grown to age 16, with a site index, based on dominant height, of 20 m (base age 15). Parameters of the population generated from the growth and yield model are presented in Table 1. Using prediction equations for the 10th and 63rd diameter percentiles of the growth model [19], Weibull parameters are recovered for the stand that are consistent with the stand-level estimate of basal area and the user-stipulated 500 trees per ha. Several parameter recovery methods for the Weibull distribution are presented in Burkhart and Tomé [12], including the use of diameter percentiles. In this case, each diameter class contains a proportion of 1/500 or 0.002 of the total trees, and the diameter class assignment is computed with the following formula by Clutter and Allison [20]:

_{i}= diameter breast height of diameter class i, and a, b, and c, are Weibull distribution parameters.

- v = total stem volume of a tree outside (including) bark in m
^{3} - w = total stem green weight outside bark in tons
- d = individual tree diameter in cm determined at breast height of 1.3 m
- h = total tree height in m

_{g}).

_{0}+ a

_{1}·(1/d), where a

_{0}and a

_{1}are estimated coefficients. Using the quadrant specific values of a

_{0}and a

_{1}, trees without observed heights were then assigned a predicted height value.

_{AC}= 4. The procedure was repeated 500 times using multiple random starts within and between iterations. For the sake of avoiding the placement of point (plot) locations too close to the long and narrow tract perimeter, there was no random perturbation of point centers from the multiple systematic grids. Point locations were at least 16 m away from the property boundary in this study location.

#### 2.2. The Application of a New Estimator for Quantifying Long-Term Production

_{AC}).

_{AC}. Rather, the estimator is a ratio that employs elements of both the Bitterlich angle count sample and the Spurr borderline factor, specifically (1−BAF

_{AC}/BLF). The new variable basal area factor estimator (BAF

_{V}) was presented by McTague [18] as:

- m = the number of “in” trees at the point (plot)
- G = the basal area in m
^{2}·ha^{−1} - BLF
_{i}= the borderline factor for tree i, or (d_{i}/R_{i})^{2}/4 - d
_{i}= diameter at 1.3 m (Dbh) in cm for tree i - R
_{i}= distance in m between the sample point and center of tree i - BAF
_{AC}= the Bitterlich angle count (point sampling) constant

_{i}is the individual tree basal area calculated as $k{d}_{i}^{2}\text{}\mathrm{with}\text{}k=\text{}\pi /40000$.

_{i}is individual tree green weight outside (including) bark. The basal area value for borderline “in” trees approaches zero, while trees close to the point center have a basal area approaching a value of 2 (BAF

_{AC}) with the new estimator. Large trees have more influence on the attributes of basal area, stem density, and volume per ha. Estimates were computed by using the BAF

_{V}and BAF

_{AC}approaches for both the 16-year old plantation in Brazil and 17-year old plantation in USA in order to compare the results obtained by both methodologies. Yield and variance statistics of repeated sampling for the stand-level attributes of basal area (G), stem density (N), and green weight per ha (W) were computed from analysis-of-variance-type calculations of the between repetitions component.

#### 2.3. Improved Compatible Estimator for Stand-Level Growth

- Z
_{AC}= change in stand green weight (t·ha^{−1}) using the additive estimator [15] of angle count sampling - w
_{ij}= green weight for tree i at time j - g
_{ij}= individual tree basal area in m^{2}·ha^{−1}for tree i at time j - m
_{j}= number of sample “in” trees at a point (plot) at time j. The number of new “in” trees at the time of remeasurement equals (m_{2}− m_{1})

_{G}represents the change in green weight (t·ha

^{−1}) using the non-additive Grosenbaugh estimator.

_{V}of Equation (5) is expressed as:

_{V}is very small for a new “in” tree at the time of remeasurement, the Z

_{V}change estimator is very efficient, as virtually all the change is attributed to the growth of survivor trees. The equations for Z

_{AC}, Z

_{G}, and Z

_{V}were used to compute the growth of stand-level green weight W. When w

_{ij}was replaced with g

_{ij}in Equations (6)–(8), stand-level growth of basal area G was computed.

#### 2.4. Alternative Measures of Individual Tree Competition

_{V}was partially comprised of the borderline factor (BLF), it is only natural to believe that it is highly correlated with the Spurr point density index [7]. The Spurr point density competition index (CI

_{PD}) requires that the (n) sample “in” trees be ordered by decreasing values of BLF

_{j}, or (d

_{j}/R

_{j})

^{2}/4, and it is expressed as:

_{V}also proves to be correlated to other distance-dependent indices. The Hegyi index (CI

_{H}) [28], as applied by Daniels and Burkhart [29], was used in a point sampling context and it is a function of size of competing trees and distance:

- d
_{i}= Dbh in cm of subject tree i - d
_{j}= Dbh in cm of competitor tree j - R
_{ij}= distance in m between subject tree i and competing tree j - n = number of trees included “in” the sample for a given BAF
_{AC}(excluding the subject tree)

_{PD}and CI

_{H}is the property that the indices predict values approaching infinity as R

_{ij}becomes increasingly smaller. This is particularly problematic for stems that are forked below breast height (1.3 m) as the distance between subject tree i and competing tree j is often less than 0.5 m.

_{V}/g) is also a function of tree size and distance (the expansion factor is used to compute the representative trees per ha of each sample tree of an angle count plot—this is unrelated to the linear expansion factor of Biging and Dobbertin [30] which is used as an adjustment for plot edge bias). Smaller trees of diameter (d) located near the angle count limiting distance, namely,$\text{}0.5d/\sqrt{BA{F}_{AC}}$, contribute less to stand weight than larger trees located near the sample point. It was hypothesized that the variable stand weight estimator, computed as the product of individual tree weight (w) and the expansion factor, might be correlated with the crown volume competition index [30]. The crown volume unweighted by distance (CVU) index was formed by projecting a 50° angle from the base of a subject tree and computing the crown volume of competing trees calculated at the point where the height angle intersects the tree bole axis. The crown is assumed to have a profile of a cone. The crown volume index is computed as:

- CVU = unweighted crown volume index
- CVa
_{j}= crown volume of competitor tree j above the point where the height angle cuts in height axis - CV
_{i}= crown volume of subject tree i - n = number of trees included “in” the sample for a given BAF
_{AC}

## 3. Results

#### 3.1. Moving the Point Center from a Probability Based Location to the Center of The Subject Tree

_{V}estimator makes it possible to assign a variable basal area factor to sample trees, based on size and distance, and efficiently estimate additive change or growth for a point or stand of interest. As demonstrated in Figure 3, when using a BAF

_{AC}= 4, the sum of the BAF

_{V}= 16.513 for the four competing trees. At each sample point, however, it is possible to further decompose the estimate of basal area and volume for the subject trees located “in” the sample and their respective competing trees. The methodology proposed here creates an absolute value of competition for the subject tree rather than a dimensionless competition index. A problem immediately surfaces, however, when attempting to quantify the contribution of basal area of the subject tree. Moving the point center from a probability based location to the center of the subject tree inflates the estimate of basal area for the subject tree and results in a bias. In other words, the BAF

_{V}value for the subject tree must increase to the maximum value (2BAF

_{AC}), resulting in a positive bias of stand basal area and volume.

_{V}value to the subject tree that is less than 2(BAF

_{AC}). The study described here attempts to quantify the new value assignment for the subject tree. The following steps are employed to compute the value for the subject tree. This new subject tree value, on average, is between the values of BAF

_{AC}and 2(BAF

_{AC}), and it should be assigned to each subject tree. It is computed with the sequence of steps below, and an example of the computations is provided in Appendix A.

- (1)
- A BAF
_{V}competition index is computed for all competitor trees surrounding every subject tree at each sample point using Equation (5). Subject trees are temporarily assigned a competition index of zero. - (2)
- The BAF
_{V}competition index is summed by subject tree. An average BAF_{V}competition index is then computed at each sample point of the probability based location. - (3)
- A difference is computed between the average variable basal area of the stand or strata (multiple sample points of the probability based design) and the average BAF
_{V}competition index for the same sample points. This average difference is then assigned as the BAF_{V}competition index for each subject tree of the stand or strata.

_{V}competition index implies knowledge of the size and location of all trees in the neighborhood of each subject tree in the probability based design. The neighborhood in this case was defined by the angle count limiting distance, namely$\text{}0.5d/\sqrt{BA{F}_{AC}}$. Edge bias was avoided in Step 1 above since all neighboring trees of subject trees were measured, irrespective of whether they are contained in the original set of the “in” trees at the probability based sample point.

#### 3.2. Sampling Simulation Statistics

_{AC}) and variable (BAF

_{V}) estimators for the stand-level attributes of basal area, green weight, and stem density are presented in Table 2 Clearly, the new BAF

_{V}estimator was more efficient in the simulated forest plantation conditions of Brazil with more uniform occupancy of the growing space. On the other hand, the advantage of the BAF

_{V}estimator for stand attributes in the United States plantation with large gaps was unclear, since it was slightly more accurate for green weight (W) and stem density (N), but less precise in terms of variance.

_{V}estimator appeared to be with permanent sample plots (PSP) and the measurement of growth. The statistical properties of growth (change) estimator, reported in Table 3, were of a repeatedly thinned plantation in Brazil and it excludes consideration of tree mortality or ingrowth (saplings) that grow into merchantable size during the remeasurement interval. Neither mortality nor ingrowth were considered as the components of growth that were responsible for the high variance of the additive (compatible) estimator. The high variance is normally attributed to the list of new “in” trees at the time of remeasurement. Since there was no change in the population value of stem density between ages 12 and 16 of the repeatedly thinned stand (see Table 1), no attempt was made to analyze the behavior of the change estimators for stem density. All estimators were accurate, but the BAF

_{V}change estimator of Equation (8) was considerably more precise than the additive estimator which uses BAF

_{AC}(Equation (6)).

_{V}estimator for the purpose of building a competition index. If the BAF

_{AC}for selecting the subject and competitor trees is larger than 3 m

^{2}·ha

^{−1}, it should be possible to measure the size and distance of all the competitor trees, irrespective of whether they were included in or out of the probability based sample. Thus, mapping all the subject and competing trees is not onerous and there is no need to employ edge correction methods for competing trees outside the plot.

#### 3.3. Correlation between BAF_{V} and Other Competition Indices

^{2}) between the BAF

_{V}estimator, Spurr CI

_{PD}, and Hegyi CI

_{H}indices for plantations in Brazil and the United States using a BAF

_{AC}= 4. The coefficient of partial determination (r

^{2}) was also reported for stand weight (W) and crown volume index of CVU. The simulated plantation conditions of Brazil have less plot to plot variability than the observed plot to plot variability in the United States. While the BAF

_{V}and W estimators were more precise for the Brazilian conditions in estimating per ha attributes of basal area and stand green weight, they were less correlated to competition indices than the United States counterpart.

#### 3.4. Assignment of an Unbiased BAF_{V} When the Subject Tree Is the Plot Center

_{V}= 2BAF

_{AC}to each subject tree led to a positive bias. For the loblolly pine plantation in southern Brazil, a simulation procedure was used to compute an unbiased BAF

_{V}value for all subject trees in the 12-year old, 40 ha forest. Although BAF

_{AC}continued to be used to determine the count of the competing “in” trees, the appropriate subject tree BAF

_{V}remained largely a function of BAF

_{AC}. Recalling Step 3 from Section 3.1, a difference was computed between the average variable basal area of the stand of the probability based design, and the average BAF

_{V}competition index for the same sample points of only the competing trees. This average difference was then applied to all subject trees of the stand or strata. An extension of this process was repeated for BAF

_{AC}values ranging from 2 to 10 m

^{2}·ha

^{−1}, as displayed in Figure 4.

_{AC}= 4, the mean unbiased BAF

_{V}for subject trees equaled 7.19 m

^{2}·ha

^{−1}. The average subject tree BAF

_{V}was not negative in any of the 500 repetitions, but it ranged from 5.47 to 9.11 m

^{2}·ha

^{−1}.

_{V}. Repeated sampling of 30 points (plots) with 500 repetitions and random starting locations was conducted for only the BAF

_{AC}= 4. The analysis revealed that the average, unbiased BAF

_{V}for the subject tree equaled 4.36 m

^{2}·ha

^{−1}. The average subject tree BAF

_{V}was not negative in any of the 500 repetitions, and it ranged from 1.26 to 7.14 m

^{2}·ha

^{−1}.

## 4. Discussion

_{V}of Equation (5). As displayed in Figure 5, the BAF

_{V}estimator is unbiased, implying that over many points, the expected values for the variable BAF estimator and the Bitterlich angle count estimator (∑BAF

_{AC}) are identical. The BAF

_{V}estimator also has desirable features, such as additivity and efficiency, when used for estimating change with permanent plots. The general consensus of the literature suggests that the Grosenbaugh growth estimator, Equation (7), is more efficient than the additive estimator, Equation (6), for most forest conditions and remeasurement intervals [25,31,32]. The drawback of the Grosenbaugh estimator is that it is non-additive (incompatible), so that volume (V

_{2}≠ [V

_{1}+ ΔV]). The disadvantage attributed to using the additive (compatible) Z

_{AC}change estimator of Van Deusen et al. [15] is manifested in terms of high variance, which is normally attributed to the list of new “in” trees at the time of remeasurement [25]. The other alternatives, Z

_{G}and Z

_{V}, are superior in providing efficient change estimates for basal area and volume per ha. If the goal is to monitor forest growth, the variable and additive BAF estimator, Z

_{V}, offers an attractive alternative to the common non-additive estimator of Z

_{G}.

_{V}estimator was advantageous because the effect of size and distance of neighboring trees is reported in absolute units of m

^{2}·ha

^{−1}(or t ha

^{−1}in the case of green weight). Most competition indices, such as Hegyi, Equation (10), or the CVU, Equation (11), do not include the contribution of the subject tree in the competition computation, since the value of the subject tree equals infinity. Of course, this drawback is not true for the BAF

_{V}estimator. In contrast, the contribution of the subject tree to the competition index based on BAF

_{V}must be included if the sum of the individual competition values are to be additive with the stand level estimate of basal area or volume.

_{AC}< 3 m

^{2}·ha

^{−1}is not recommended. It was desirable to keep the total sample size of subject and competing trees manageable by using a BAF

_{AC}≥ 3. One recent investigation arrived at the conclusion that the Bitterlich angle count method was not the best method for selecting competitor trees [6]. Here, we contend that Bitterlich method should not be used exclusively for just selecting competitor trees, but rather the point sampling expansion factors should also be employed to scaling the competition indices up to a per ha level. If the expansion factor of (BAF

_{V}/g) is used, the competition indices are both spatially explicit and can express the competition in absolute values at a per ha level.

_{V}estimator for measuring competition is that it is biased for measuring the competition contribution of the subject tree. This paper presents a methodology for arriving at a stand level value for BAF

_{V}for the subject tree that removed the bias. The proposed method then assigns the same BAF

_{V}value to all subject trees over all points (plots) in the stand. Compatibility or additivity is achieved at the stand level, but not at the point or plot level. Compatibility is mathematically possible at the point (plot) level, however, it may result in either the negative assignment of BAF

_{V}for the subject tree, or the need to adjust the values of both subject and competing trees. Clearly, the drawbacks of achieving compatibility at the point (plot) level outweigh the possible benefits.

^{2}·ha

^{−1}(7.19 m

^{2}·ha

^{−1}vs. 4.36 m

^{2}·ha

^{−1}) between the assignment of the average unbiased BAF

_{V}value for the subject tree, using a BAF

_{AC}= 4, when comparing southeastern United States and Brazil. Clearly the variability in the simulated Brazilian plantation was less pronounced than the observed conditions in southeastern United States, with large holes in the forest canopy and frequent null tallies at some points (plots) in the latter. In hindsight, the study would have benefited from a larger stem-mapped forest in the United States, or with a more equal distribution of forest canopy holes between the center and periphery of the stand. Without question, the assignment of 2(BAF

_{AC}) to each subject tree results in a positive bias, irrespective of the forest structure and conditions.

_{V}estimator is perfectly suited for measuring growth. In addition, it is well known that angle count samples are inexpensive to install and maintain. With a selection of a BAF

_{AC}≥ 3, it should be physically and economically feasible to measure all competitor trees and obviate the requirement for edge correction. A difficulty can arise, however, when using a BAF

_{AC}≥ 3, if the forest is subjected to intermediate harvests that involve strips or mechanical row thinning. Depending on the width of the harvest cut, a large proportion of the subject and competing trees of the point can be eliminated, compromising the utility of the remeasured plot over the next measurement interval. This is particularly true for forest stands where the quadratic mean diameter (d

_{g}) is less than 15 cm and the equivalent plot size is rather small. As trees grow larger in size and the effective plot size increases, the abrupt change in sample size associated with cutting diminishes.

## 5. Conclusions

## Appendix A

_{V}competition index for the subject trees. For the purpose of simplicity, the procedure is demonstrated using a single angle count sample of the probability based sample. On a practical basis, however, it is recommended that the BAF

_{V}competition index should be computed for the subject trees over all probability based sample points (plots) in the stand or strata. Table A1 provides the computations that follow the three steps contained in Section 3.1 for assigning a BAF

_{V}competition index to the subject trees. Figure A1 displays the spatial configuration of subject and competition trees of the point. As the estimate of stand basal area is G = 15. 24 m

^{2}·ha

^{−1}and the average BAF

_{V}without the subject tree equals 12.20, the difference of 3.04 represents the appropriate assignment for the BAF

_{V}competition index for all subject trees to achieve compatibility between the stand-level estimate of basal area and the individual tree competition indices.

**Table A1.**Computations for achieving compatibility between the stand-level estimate of basal area and the assignment of individual-tree competition indices. BAF

_{AC}= 4.

Subject Tree (st) | Distance (R) from Point Center (m) | Dbh, d (cm) | BAF_{V} | Competition Tree | Distance (R) from Subject Tree (m) | Dbh, d (cm) | BAF_{V} |
---|---|---|---|---|---|---|---|

4 | 2.67 | 19.1 | 5.50 | ||||

3 | 7.33 | 36.1 | 2.72 | ||||

5 | 3.21 | 17.0 | 3.44 | ||||

6 | 6.11 | 31.9 | 3.30 | ||||

10 | 6.64 | 42.5 | 4.88 | ||||

∑BAF_{V} (w/o st) | 14.34 | ||||||

5 | 4.20 | 17.0 | 0.19 | ||||

4 | 3.20 | 19.1 | 4.41 | ||||

6 | 5.50 | 31.9 | 4.20 | ||||

10 | 8.93 | 42.5 | 2.35 | ||||

∑BAF_{V} (w/o st) | 10.95 | ||||||

6 | 6.26 | 31.8 | 3.04 | ||||

7 | 2.44 | 21.2 | 6.30 | ||||

8 | 5.8 | 25.5 | 1.38 | ||||

10 | 6.11 | 42.5 | 5.35 | ||||

∑BAF_{V} (w/o st) | 13.04 | ||||||

10 | 4.58 | 42.5 | 6.51 | ||||

1 | 3.82 | 19.1 | 2.88 | ||||

2 | 2.44 | 12.7 | 3.28 | ||||

6 | 6.11 | 31.9 | 3.30 | ||||

9 | 3.97 | 17.0 | 1.02 | ||||

∑BAF_{V} (w/o st) | 10.48 | ||||||

G = | 15.24 | ave. BAF_{V} (w/o st) | 12.20 |

**Figure A1.**Map of subject trees (

**blue**) and competitor trees (

**green**) for a sample point. Note that subject trees can be competitor trees of other subject trees.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Weiskittel, A.R.; Hann, D.W.; Kershaw, J.A., Jr.; Vanclay, J.K. Forest Growth and Yield Modeling; John Wiley & Sons: Chichester, UK, 2011. [Google Scholar]
- Pretzsch, H. Forest Dynamics, Growth and Yield: From Measurement to Model; Springer: Berlin, Germany, 2009. [Google Scholar]
- Daniels, R.F. Notes: Simple competition indices and their correlation with annual loblolly pine tree growth. For. Sci.
**1976**, 22, 454–456. [Google Scholar] - Pedersen, R.Ø.; Nӕsset, E.; Gobakken, T.; Bollandsås, O.M. On the evaluation of competition indices—The problem of overlapping samples. For. Ecol. Manag.
**2013**, 310, 120–133. [Google Scholar] [CrossRef] - Soares, P.; Tomé, M. Distance-dependent competition measures for eucalyptus plantations in Portugal. Ann. For. Sci.
**1999**, 5, 307–319. [Google Scholar] [CrossRef] - Maleki, K.; Kiviste, A.; Korjus, H. Analysis of individual tree competition effect on diameter growth of silver birch in Estonia. For. Syst.
**2015**, 24, e023. [Google Scholar] - Spurr, S.H. A measure of point density. For. Sci.
**1962**, 8, 85–96. [Google Scholar] - Bitterlich, W. Die Winkelzählprobe. Allem. For. Holzwirtsch. Ztg.
**1948**, 59, 4–5. [Google Scholar] [CrossRef] - Tomé, M.; Burkhart, H.E. Distance-dependent competition measures for predicting growth of individual trees. For. Sci.
**1989**, 35, 816–831. [Google Scholar] - Rouvinen, S.; Kuuluvainen, T. Structure and asymmetry of tree crowns in relation to local competition in a natural mature Scots pine forest. Can. J. For. Res.
**1997**, 27, 890–902. [Google Scholar] [CrossRef] - Bitterlich, W. The Relascope Idea: Relative Measurements in Forestry; Commonwealth Agricultural Bureaux: Farnham Royal, UK, 1984. [Google Scholar]
- Burkhart, H.E.; Tomé, M. Modeling Forest Trees and Stands; Springer: Dordrecht, The Netherlands, 2012. [Google Scholar]
- Husch, B.; Miller, C.I.; Beers, T.W. Forest Mensuration, 3rd ed.; John Wiley & Sons: New York, NY, USA, 1982. [Google Scholar]
- Furnival, G. Forest sampling—Past performance and future expectations. In Forest Resource Inventories Workshop Proceedings; Frayer, W.E., Ed.; Colorado State University: Fort Collins, CO, USA, 1979; Volume 1, pp. 320–326. [Google Scholar]
- Van Deusen, P.C.; Dell, T.R.; Thomas, C.E. Volume growth estimation for permanent horizontal points. For. Sci.
**1986**, 32, 415–422. [Google Scholar] - Grosenbaugh, L.R. Point-Sampling and Line-Sampling: Probability Theory, Geometric Implications, Synthesis; USDA Forest Service, Southern Forest Experiment Station: New Orleans, LA, USA, 1958.
- Green, E.J. Forest growth with point sampling data. Conserv. Biol.
**1992**, 6, 296–297. [Google Scholar] [CrossRef] - McTague, J.P. New and composite point sampling estimates. Can. J. For. Res.
**2010**, 40, 2234–2242. [Google Scholar] [CrossRef] - McTague, J.P.; Bailey, R.L. Compatible basal area and diameter distribution models for thinned loblolly pine plantations in Santa Catarina, Brazil. For. Sci.
**1987**, 33, 43–51. [Google Scholar] - Clutter, J.L.; Allison, B.J. A growth and yield model for Pinus radiata in New Zealand. In Growth Models for Tree and Stand Simulation; Fries, J., Ed.; Research Note No. 30; Royal College of Forestry, Department of Forest Yield Research: Stockholm, Sweden, 1974; pp. 136–170. [Google Scholar]
- Padoin, V.; Finger, C.A.G. Relações entre as dimensões da copa e a altura das árvores dominantes em povoamentos de Pinus taeda L. Ciênc. Florest.
**2010**, 20, 95–100. [Google Scholar] [CrossRef] - Harrison, W.M.; Borders, B.E. Yield Prediction and Growth Projection for Site-Prepared Loblolly Pine Plantations in the Carolinas, Georgia, Alabama, and Florida; PMRC Technical Report. 1996-1; Warnell School of Forest Resources, Plantation Management Research Cooperative, University of Georgia: Athens, GA, USA, 1996. [Google Scholar]
- Bechtold, W.A. Crown-diameter prediction models for 87 species of stand-grown trees in the eastern United States. South. J. Appl. For.
**2003**, 27, 269–278. [Google Scholar] - Dyer, M.E.; Burkhart, H.E. Compatible crown ratio and crown height models. Can. J. For. Res.
**1987**, 17, 572–574. [Google Scholar] [CrossRef] - Hradetzky, J. Concerning the precision of growth estimation using permanent horizontal point samples. For. Ecol. Manag.
**1995**, 71, 203–210. [Google Scholar] [CrossRef] - McTague, J.P. The use of composite estimators for estimating forest biomass and growth from permanent sample plots established by the angle count method. In Proceedings of the 2013 Joint Statistical Meetings, Montréal, QC, Canada, 3–8 August 2013; American Statistical Association: Washington, DC, USA, 2015; pp. 4663–4677. [Google Scholar]
- Beers, T.W.; Miller, C.I. Point Sampling: Research Results Theory and Applications; Purdue University Agricultural Experiment Station: Lafayette, IN, USA, 1964; No. 786. [Google Scholar]
- Hegyi, F. A simulation model for managing Jack-pine stands. In Growth Models for Tree and Stand Simulation; Fries, J., Ed.; Research Note No. 30; Royal College of Forestry, Department of Forest Yield Research: Stockholm, Sweden, 1974; pp. 74–90. [Google Scholar]
- Daniels, R.F.; Burkhart, H.E. Simulation of Individual Tree Growth and Stand Development in Managed Loblolly Pine Plantations; Publication No. FWS-5-75; Division of Forestry and Wildlife Resources, Virginia Polytechnic Institute and State University: Blacksburg, VA, USA, 1975.
- Biging, G.S.; Dobbertin, M. A comparison of distance-dependent competition measures for height and basal area growth of individual conifer trees. For. Sci.
**1992**, 38, 695–720. [Google Scholar] - Gregoire, T.G. Estimation of forest growth from successive surveys. For. Ecol. Manag.
**1993**, 56, 267–278. [Google Scholar] [CrossRef] - Thérien, G. Relative efficiency of point sampling change estimators. Math. Comput. For. Nat. Res. Sci.
**2011**, 3, 64–72. [Google Scholar]

**Figure 1.**An angle stick gauge of a predetermined length r (BAF

_{AC}) from the peep sight and a crossarm of predetermined width l is used to determine the sample size of neighbors in the competition index for the subject tree. Neighbor 1 is a competitor tree because it subtends an angle that is larger the critical angle defined by the basal area factor of angle count sampling (BAF

_{AC}). Conversely, Neighbor 2 is not included in the sample and is considered a non-competitor. BAF

_{AC}, angle count basal area factor; BLF, borderline factor or (d/R)

^{2}/4. (Adapted from [12].)

**Figure 2.**Aerial images for the Southeastern United States study site. Panel (

**a**) displays a recently thinned, 17-year old plantation, containing planting rows with curvature and numerous switches in planting direction; Panel (

**b**) is the 13.1 ha rectangle from the center of the stand and contains 135 quadrants with dimensions of 31.15 × 31.15 m. Individual trees were mapped and depicted with white dots.

**Figure 3.**Conceptual figure demonstrating calculations of new estimator. Each competing tree is assigned a variable basal area factor (BAF

_{V}) depending upon its size and distance to the subject tree.

**Figure 4.**Box plot of the subject tree variable basal area factor (BAF

_{V}) and the traditional angle count factor (BAF

_{AC}) for a loblolly pine plantation in Brazil. Each open diamond symbol represents the average value of 500 iterations.

**Figure 5.**Employing the example from Brazil, the population value of the stand at age 12 for total basal area (G; m

^{2}·ha

^{−1}) is equal to 28.8. Both estimators, namely BAF

_{AC}and BAF

_{V}, are unbiased for any basal area factor, however, the BAF

_{V}estimator is somewhat more precise when BAF

_{AC}< 4.

**Table 1.**Population parameters of thinned plantations of loblolly pine in Brazil and the United States used in this analysis.

Attribute | Measurement 1 | Measurement 2 | Change |
---|---|---|---|

Santa Catarina, Brazil | |||

Total age (years) | 12 | 16 | 4 years |

Stem density (N) | 500 | 500 | 0 |

Forest stand basal area (G; m^{2}·ha^{−1}) | 28.8 | 39.1 | 10.3 |

Total green weight outside bark (W; t·ha^{−1}) | 195.2 | 335.5 | 140.3 |

Mean top height (H; m) | 17.0 | 20.9 | 3.9 |

Quadratic mean diameter (d_{g}; cm) | 27.1 | 31.6 | 4.5 |

Weibull location (a) parameter | 19.13 | 22.11 | - |

Weibull scale (b) parameter | 8.709 | 10.257 | - |

Weibull shape (c) parameter | 2.241 | 2.015 | - |

Southeastern United States | |||

Total age (years) | 17 | - | - |

Stem density (N) | 330.5 | - | - |

Forest stand basal area (G; m^{2}·ha^{−1}) | 14.13 | - | - |

Total green weight outside bark (W; t·ha^{−1}) | 100.0 | - | - |

Quadratic mean diameter (d_{g}; cm) | 23.3 | - | - |

**Table 2.**Estimates and efficiencies of the point sampling estimators obtained from repeated sampling using a BAF

_{AC}= 4 for basal area, stem density, and green weight per ha.

Age, Location, andEstimator | Basal Area (G; m^{2}·ha^{−1}) $\sum \mathit{B}\mathit{A}\mathit{F}$ | Stem Density (N) $\sum \mathit{B}\mathit{A}\mathit{F}\frac{1}{{\mathit{g}}_{\mathit{i}}}$ | Green Weight (W; t·ha^{−1}) $\sum \mathit{B}\mathit{A}\mathit{F}\frac{{\mathit{w}}_{\mathit{i}}}{{\mathit{g}}_{\mathit{i}}}$ | |||
---|---|---|---|---|---|---|

Mean | Variance of Mean | Mean | Variance of Mean | Mean | Variance of Mean | |

Age 16—Brazil | ||||||

Conventional BAF_{AC} | 39.1 | 0.8 | 500.2 | 158 | 335.9 | 63 |

Equation (5) BAF_{V} | 39.2 | 0.6 | 500.7 | 98 | 336.2 | 45 |

Age 17—USA | ||||||

Conventional BAF_{AC} | 14.0 | 1.7 | 357.8 | 779 | 109.3 | 63 |

Equation (5) BAF_{V} | 14.1 | 1.9 | 325.3 | 1027 | 99.5 | 107 |

_{i}is the individual tree basal area.

**Table 3.**Estimates and efficiencies of the point sampling change estimators obtained for basal area and green weight per ha in Brazil.

Estimator | Equation and Source | Change in Basal Area (ΔG; m^{2}·ha^{−1}) | Change in Green Weight (ΔW; t·ha^{−1}) | ||
---|---|---|---|---|---|

Mean | Variance of Mean | Mean | Variance of Mean | ||

Z_{AC} | Equation (6), [15] | 10.3 | 0.81 | 140.0 | 52.9 |

Z_{G} | Equation (7), [16] | 10.3 | 0.10 | 140.5 | 16.8 |

Z_{V} | Equation (8), [29] | 10.3 | 0.18 | 140.3 | 13.4 |

_{AC}, change in stand green weight (t·ha

^{−1}) using the additive estimator of angle count sampling; Z

_{G}, the change in green weight (t·ha

^{−1}) using the non-additive Grosenbaugh estimator; Z

_{V}, change estimator using BAF

_{V}.

**Table 4.**Coefficients of partial determination for selected competition indices and the BAF

_{V}and W estimators. The coefficient was defined as ${r}_{y3.12}^{2}=1-\frac{SSE\text{}\left({X}_{1},\text{}{X}_{2},\text{}{X}_{3}\right)}{SSE\left({X}_{1},\text{}{X}_{2}\right)}$ , where SSE is the sum of squares of error, X

_{1}is a class variable denoting iteration, X

_{2}is a class variable denoting plot, and X

_{3}is either BAF

_{V}or W. y denotes the dependent variable of CI

_{PD}, CI

_{H}, or CVU. The coefficient of partial determination between y and X

_{3}, given that X

_{1}and X

_{2}are in the model is expressed as “${r}_{y3.12}^{2}$”. SSE(X

_{1}, X

_{2}, X

_{3}) is computed with the full model containing X

_{1}, X

_{2}, and X

_{3}as independent variables, while SSE(X

_{1}, X

_{2}) is the variation associated with the reduced model containing X

_{1}, and X

_{2}as independent variables.

Country | Indices | Coefficient of Partial Determination (r^{2}) |
---|---|---|

Brazil | BAF_{V} and CI_{PD} | 0.718 |

BAF_{V} and CI_{H} | 0.582 | |

W and CVU | 0.212 | |

United States | BAF_{V} and CI_{PD} | 0.903 |

BAF_{V} and CI_{H} | 0.724 | |

W and CVU | 0.646 |

_{PD}and CI

_{H}are the Spurr point density and Hegyi competition indices respectively. CVU is the unweighted crown volume index. BAF

_{V}and W are the proposed competition indices for basal area and weight.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

McTague, J.P.; Weiskittel, A.R. Individual-Tree Competition Indices and Improved Compatibility with Stand-Level Estimates of Stem Density and Long-Term Production. *Forests* **2016**, *7*, 238.
https://doi.org/10.3390/f7100238

**AMA Style**

McTague JP, Weiskittel AR. Individual-Tree Competition Indices and Improved Compatibility with Stand-Level Estimates of Stem Density and Long-Term Production. *Forests*. 2016; 7(10):238.
https://doi.org/10.3390/f7100238

**Chicago/Turabian Style**

McTague, John Paul, and Aaron R. Weiskittel. 2016. "Individual-Tree Competition Indices and Improved Compatibility with Stand-Level Estimates of Stem Density and Long-Term Production" *Forests* 7, no. 10: 238.
https://doi.org/10.3390/f7100238