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Longleaf pine (
Longleaf pine (
In order to improve stand management planning, researchers, managers and landowners need reliable information about stand dynamics and development. While a number of longterm experimental and permanent plot datasets exist for the species, these data, for the most part, have not been summarized in a comprehensive manner. As forest management decisions are based on information about current and future resource conditions, forest growth and yield modeling plays an important role by quantifying and summarizing relationships observed in field studies, and by providing stand projections under alternative management scenarios. Wholestandlevel growth and yield models predict future yields as a function of previous standlevel attributes such as age, stand density and site quality [
A number of models have been produced that predict elements of longleaf pine plantation stand dynamics [
The objective of this study was to develop a standlevel growth and yield model system for thinned and unthinned longleaf pine plantations, using a longterm (>40 years) dataset measured and maintained by the U.S. Forest Service. A system of equations was developed to summarize the dynamics observed in the extensive, longterm dataset, and to provide a tool to predict and project stand growth and yield, including merchantable volume breakdown functions.
The dataset used to estimate growth and yield parameters comes from 267 permanent plots measured and maintained by the U.S. Forest Service’s Laboratory at Pineville, LA [
Location of 267 permanent plots measured within the Western Gulf Coastal Plain longleaf pine natural distribution range.
For each tree, stem diameter outside bark at 1.37 m height (dbh, cm) was measured to the nearest 0.25 cm and total tree height (H, m) was measured to the nearest 0.3 m on a subsample of 4 to 15 trees per plot. Using the model proposed by Quicke and Meldahl [
From the complete dataset, 30 plots were randomly selected and removed to use for model evaluation and the rest (
Summary statistics and stand characteristics for 267 permanent longleaf pine plots measured (thinned and unthinned).
Variable  Model fitting dataset (237 plots)  Model evaluation dataset (30 plots)  

Mean  SD  Minimum  Maximum 

Mean  SD  Minimum  Maximum 


Age  35.9  15.7  7.0  73.0  725  36.8  16.3  7.0  73.0  140 
dbh  21.6  8.3  6.8  44.1  725  21.6  7.8  6.8  42.4  140 
H  18.6  5.3  5.3  30.1  725  19.3  5.8  5.4  30.5  140 
N  865  504  99  2849  725  982  548  198  2422  140 
BA  27.0  11.3  6.6  62.9  725  31.5  13.1  6.8  65.9  140 
Dq  22.6  8.4  7.0  44.3  725  22.6  8.0  7.1  42.8  140 
SDI  566  222  129  1222  725  655  250  144  1287  140 
H_{dom}  20.6  5.6  6.7  32.0  725  21.5  6.1  6.4  32.8  140 
SI  25.8  1.9  19.6  30.8  173 *  26.4  1.6  22.1  29.3  21 * 
VOL_{OB}  274.6  141.2  46.7  688.6  725  320.7  161.4  55.0  701.3  140 
Age: stand age (year); dbh: arithmetic mean diameter outside bark at breast height (cm); H: total height (m); N: trees per hectare (trees ha^{−1}); BA: basal area (m^{2}·ha^{−1}); Dq: quadratic mean diameter (cm); SDI: Reineke’s stand density index (trees ha^{−1}); H_{dom}: height of dominant and codominant trees (m); SI: site index (m); VOL_{OB}: total stem volume outsidebark (m^{3}·ha^{−1}); SD: standard deviation;
For all trees with H measurement (28,083 observations) a form factor F = H/dbh (m cm^{−1}) was calculated. In order to eliminate broken and malformed individuals, trees with F less than 0.54 m cm^{−1} and trees with F greater than 13.5 m·cm^{−1} were discarded for H_{dom} determination (5.4% of H observations). Plots with less than four trees with H measured were also not considered for H_{dom} fitting (7.9% of total plots).
The distribution of total observations (used for model fitting and model evaluation) by age, SI and surviving trees (N, ha^{−1}) is shown in
Distribution of observations by age, site index and surviving density for 267 permanent longleaf pine plots measured (thinned and unthinned).
Surviving Density Class (ha^{−1})  Stand Age Class (yrs.)  

7–20  20–40  40–60  60–73  Total  
99–500  8  66  105  61  240 
500–1000  51  112  91  17  271 
1000–1500  69  153  27    249 
1500–2000  36  49  2    87 
2000–2849  10  8      18 


19–22  4  45  12  4  35 
22–25  46  78  46  21  191 
25–28  49  217  132  49  447 
28–31  75  78  35  4  192 






*: SI reported for plots not measured at age 50 yrs. was estimated with model presented in this study.
Only 2% of plots had N > 2000 ha^{−1}, while 28% of data points had N < 500 ha^{−1}. About 29% of data had N between 1000 and 1500 ha^{−1}. Most stands (52%) had SI between 25 and 28 m, and only 4% of data had SI < 22 m. About 22% of stands had SI > 28 m. In term of age distribution, about 20% of data had age < 20 yrs. and 9% had age > 60 yrs.
Data from all unthinned plots and from thinned plots prior to thinning were used to estimate survival, H_{dom}, stand basal area and stand volume (inside and outside bark) parameters.
Following the guide curve method to produce an anamorphic model [
This equation can be inverted to determine SI if stand age and H_{dom} are known. This anamorphic model has the assumption that the shape of the heightage curve is independent of SI, and differences between any two curves are proportional to the ratio of their SI’s [
A negativeexponential survival model that includes H_{dom} was used to estimate survival using a modified version of the model proposed by DieguezAranda
The following generic equation, proposed by Borders [
The equations reported by GonzalezBenecke
As the effects of thinning on survival and H_{dom} are small for southern pines (Westfall and Burkhart, 2001; Sharma
From Equation 7, when BA_{t} equals BA_{u} with the same number of trees, the CI is zero. Similarly, when BA_{t} is less than BA_{u} (that is the general case in operational thinnings), the CI is larger than zero, but approaching zero as stand ages, as BA_{t} will converge to BA_{u} [
After combining Equations 5 and 6, BA of a thinned stand is estimated using the projected CI as:
For each tree in the fitting dataset, merchantable stem volume (both outside and inside bark), from the stump to any top diameter, was estimated using the equations reported by GonzalezBenecke
The merchantable volume yield breakdown at the stand level function was determined following Amateis
The predictive ability of the fitted models for N (Equation 4), H_{dom} (Equation 2), BA of unthinned stands (Equation 6), VOL of unthinned stands (Equation 8) and BA of thinned stands (Equation 11) was assessed with the 30 plot evaluation dataset. In the case of VOL_{m} (Equation 12), two combinations of
Four measures of accuracy were used to evaluate the goodnessoffit between observed and predicted (simulated) values for each variable originated from the dataset obtained in the model evaluation: (i) mean absolute error (MAE); (ii) root mean square error (RMSE); (iii) mean bias error (Bias); and (iv) coefficient of determination (R^{2}) [
The system of equations was also compared against other models reported in the literature for longleaf pine plantations using the same model evaluation dataset indicated above. The models compared were: (i) survival equations reported by Lohrey and Bailey [
An overall evaluation of the model was carried out for unthinned plots of the validation dataset. On each plot, for known initial stand age, N and SI, stand BA and VOL_{OB} were estimated on the same ages where they were originally measured by using the final equations fitted to estimate N, H_{dom}, BA and VOL_{OB}. The same four measures of accuracy described previously were used to assess the agreement between observed and predicted values.
All of the summary, model fitting and model evaluation statistics were obtained using SAS 9.3 (SAS Inc., Cary, NC, USA) [
The system of equations developed was used to predict stand growth of unthinned and thinned (3 thinnings, removing 33% of living trees at ages 30, 40 and 50 yrs.) longleaf pine stands growing at sites with two different SI’s: 20 and 30 m. The initial planting density used was 1400 trees ha^{−1}, the survival after the first year was assumed to be 95%, and the simulation length was 70 yrs. Here it was assumed that the percentage of removed trees was the same as the percentage removal of BA during thinning.
The parameter estimates for the growth and yield predictive and projective equations (Equations 2, 4, 6, 8, 10 and 12) for longleaf pine plantations growing in Western Gulf Coastal Plain U.S. are reported in
Parameter estimates for the model that projects dominant height (Equation 2) are shown in
The survival model (Equation 4) was dependent on stand age and H_{dom}. The performance of the N model for the range of SI present on the dataset used for model fitting (
After applying the stepwise procedure and checking variance inflation factors (VIF), the final selected model that predicts BA (Equation 6) was only dependent on N and H_{dom} (
The final model that predicts stand volume (Equation 8), after the stepwise variable selection procedure and VIF checking, was dependent on N, BA, ln(BA)/Age and SI (
Parameter estimates and fit statistics of Western Gulf Coastal Plain U.S. longleaf pine plantation growth and yield equations.
Model 

Parameter  Parameterestimate  Approx.SE  Approx. Pr > F  VIF  R^{2}  RMSE  CV% 

569 

−0.0369815  0.0015463  <0.0001  n.a.  0.998  0.87  4.1  

1.2928702  0.0454849  <0.0001  n.a.  
622 

−0.0015002  0.0006992  0.0324  n.a.  0.997  52.89  6.9  

0.8635401  0.1000509  <0.0001  n.a.  
725  −4.6484039  0.0736689  <0.0001  0  0.944  1.12  3.6  

0.4452486  0.0064583  <0.0001  1.16  

1.6526307  0.0155728  <0.0001  1.16  
569  3.1110579  0.0809271  <0.0001  0  0.997  0.04  0.72  

−0.1406022  0.0045948  <0.0001  4.41  

1.1826310  0.0040024  <0.0001  2.48  

−2.4435259  0.0989071  <0.0001  4.02  

−0.0782880  0.0265719  0.0033  1.49  
569  3.0888853  0.1026120  <0.0001  0  0.996  0.05  1.0  

−0.1943861  0.0058271  <0.0001  4.41  

1.2580580  0.0050738  <0.0001  2.48  

−3.1281571  0.1254092  <0.0001  4.02  

−0.098259  0.0336921  0.0037  1.49  
292 

−1.5476196  0.1881092  <0.0001  n.a.  0.874  0.05  96.5  
21,541 

−1.0385828  0.0026438  <0.0001  n.a.  0.990  0.07  11.1  

4.2526170  0.0147436  <0.0001  n.a.  

−0.6266850  0.0596972  <0.0001  n.a.  

−0.1246646  0.0185442  <0.0001  n.a.  

9.1649608  0.1878172  <0.0001  n.a.  
21,541 

−1.0537628  0.0027184  <0.0001  n.a.  0.990  0.07  11.2  

4.2527499  0.0148697  <0.0001  n.a.  

−0.6545719  0.0641831  <0.0001  n.a.  

−0.1365633  0.0191092  <0.0001  n.a.  

9.3108306  0.1971518  <0.0001  n.a. 
H_{dom}: average total height (m) of dominant and codominant trees; SI: site index (m); N
The model that projects the time trend of CI after thinning (Equation 10) was dependent on stand age (
Parameter estimates for merchantable volume yield breakdown function (Equation 12) for both, outside (VOL_{m}_{OB}, m^{3}·ha^{−1}) and inside (VOL_{m}_{IB}, m^{3}·ha^{−1}) bark total volume yield, are shown in
There was a good agreement between predicted and observed values of N (
Validation of dominant height (Hdom) (
If residuals are expressed as a percentage of the observed value, maximum absolute residuals observed represent about 17% and 16% of observed N and H_{dom}, respectively. Residuals for predicting the BA model were larger, with maximum residuals of about 30% of observed BA, but centered around zero. There was no noticeable trend in residuals with observed values (
A growth model to project, or update, BA (BA_{j}) when some stand measurements are available, including current BA (BA
As expected, this projecting growth model improved the estimations of BA, reducing the residuals as compared with BA predicting model (
There was good agreement between predicted and observed values for VOL outside and inside bark (
For the two combinations of
Validation of total stem volume outside (VOL_{OB}) (
All model performance tests showed that N, H_{dom}, BA, BA_{t}, VOL and VOL_{m} estimations agreed well with measured values (
Summary of model evaluation statistics for N, H_{dom}, BA, BA_{t}, VOL, and VOL_{m} estimations based on 30 plots.
Variable 

MAE  RMSE  Bias  R^{2}  

N  938  929  120  40.3 (4.3)  53.1 (5.7)  −9.46 (−1.0)  0.991 
H_{dom}  23.0  22.8  111  0.6 (2.7)  0.8 (3.6)  −0.20 (−0.9)  0.978 
BA  31.5  30.8  140  3.1 (10.0)  4.2 (13.5)  −0.65 (−2.1)  0.898 
BA_{j}  33.1  33.3  120  1.9 (5.8)  2.6 (7.8)  0.45 (1.4)  0.970 
VOL_{OB}  340.4  341.1  115  11.9 (3.5)  16.4 (4.8)  0.61 (0.2)  0.990 
VOL_{IB}  269.8  270.9  115  12.3 (4.6)  17.2 (6.4)  1.06 (0.4)  0.984 
BA_{t}  28.5  27.9  52  0.9 (3.1)  1.1 (3.8)  −0.65 (−2.3)  0.984 
VOL_{m}_{OBt = 10, d = 20}  137.1  138.3  70  2.5 (1.9)  4.0 (1.9)  1.25 (0.9)  0.999 
VOL_{m}_{IBt = 10, d = 20}  109.9  110.8  70  1.9 (1.7)  3.0 (1.7)  0.94 (0.9)  0.999 
VOL_{m}_{OBt = 20, d = 30}  140.4  141.5  20  7.9 (5.6)  9.0 (5.6)  1.14 (0.8)  0.998 
VOL_{m}_{IBt = 20, d = 30}  115.0  114.4  20  6.3 (5.5)  7.2 (5.5)  −0.63 (−0.5)  0.998 
N: trees per hectare (ha^{−1}); H_{dom}: average total height of dominant and codominant trees (m); BA: predicted basal area of unthinned stands (m^{2}·ha^{−1}); BA_{j}: projected basal area of unthinned stands (m^{2}·ha^{−1}); VOL: total stem volume (m^{3}·ha^{−1}); BA_{t}: basal area of thinned stands (m^{2}·ha^{−1}); VOL_{m}: merchantable stem volume for trees
When tested on the dataset used for model evaluation, predicted values of the models proposed in this study for N, H_{dom} and VOL_{OB} are within the range of variation of the estimations using other published growth and yield models. The effects of stand age on survival, H_{dom}, and VOL_{OB} were predicted using several models for longleaf pine (
Similarly, the model presented in this study underpredicted VOL_{OB} by about 30 m^{3}·ha^{−1} (or around 7%) for older stands (
Mean bias and RMSE of the models presented in this study and reported in literature to predict survival (
Examples of merchantable yield breakdown function of VOL_{OB} estimations for
Comparison of merchantable volume yield breakdown functions published for
For example, for sawtimber, defined as stem volume of trees with dbh larger than 29.2 cm outside bark (threshold dbh limit) to a top diameter of 20.3 cm outside bark (merchantability limit), when Dq was smaller than 20 cm there was no sawtimber volume production, but when Dq was 30 cm, sawtimber yield was about 55, 61 and 52% (N = 100 trees ha^{−1}) or 57, 67 and 73% (N = 1000 trees ha^{−1}) for
The overall test of the model indicated that, if only initial (
Overall simulation validation of survival (N) (
If initial stand age, N and SI are known, the overall test of the model system indicated that projections of N, H_{dom} and predictions of BA and VOL_{OB} for less than ~40 yrs. simulation length presented a bias that ranged between −7% and 10% (
Summary of overall model evaluation statistics for N, H_{dom}, BA and VOL_{OB} estimations using different reference age for SI for different simulation lengths.
Variable  Simulation length (yrs.) 

MAE(%)  RMSE(%)  Bias(%)  R^{2}  

N  0–20  818.5  818.3  339  8.3%  12.4%  0.0%  0.959 
21–40  541.1  560.9  172  21.3%  30.5%  3.5%  0.884  
All  20.9  20.9  339  11.6%  17.6%  0.9%  0.934  
H_{dom}  0–20  27.2  27.2  172  3.3%  4.2%  −0.1%  0.960 
21–40  23.0  23.0  511  1.7%  2.1%  0.0%  0.948  
All  36.1  34.9  172  2.6%  3.4%  −0.1%  0.974  
BA  0–20  30.6  29.7  511  9.8%  13.1%  −2.7%  0.887 
21–40  273.9  273.0  339  15.9%  19.1%  −3.5%  0.747  
All  333.6  320.1  511  12.2%  16.2%  −3.0%  0.837  
VOL_{OB}  0–20  818.5  818.3  339  10.4%  13.6%  −0.3%  0.895 
21–40  541.1  560.9  172  19.6%  23.2%  −8.7%  0.557  
All  20.9  20.9  339  13.2%  18.3%  −4.2%  0.839 
N: trees per hectare (ha^{−1}); H_{dom}: average total height of dominant and codominat trees (m); BA: stand basal area (m^{2}·ha^{−1}); VOL_{OB}: total stem volume outside bark (m^{3}·ha^{−1});
An example of model behavior for a hypothetical longleaf stands planted with 1400 trees ha^{−1} is shown in
Example of model outputs. Simulation of survival (N, trees ha^{−1}): (
Bringing existing longleaf pine stands under management and restoring longleaf pine stands from degraded or otherwise converted forest stands is a priority for a number of land management entities in the southeastern U.S. [
All choices of model structure involve compromise. Wholestand level models, as the one presented here, provide reliable prediction of stand variables, such as BA and N; on the other hand, they do not provide the level of detail that individualtree level models produce, which could allow for more flexibility in modeling silvicultural practices. However, individualtree models typically are unreliable in prediction of cumulated stand information, and often have issues with propagation of errors. In this study, we opted to fit a stand level model to be used as a baseline, and in future work, we will consider incorporating individualtree level information.
Site index is the most widely used measure of forest productivity, particularly in plantations. Base age selection for SI can have significant implications for the accuracy of estimations, as bias increases as the stand age is further from the base age [
In relation to the survival equations, the best model fitted was dependent on stand age and dominant height (a measure of site quality). Other models also incorporate the effect of site quality on survival, such as the models reported by Lauer and Kush [
Other models that included SI [
The model that presdict BA for unthinned stands was only dependent on N and H_{dom}. Other models reported for southern pine species include other simple or composite variables such as SI, Age, N/Age and H_{dom}/Age as well. In this study, all of these variables were significant and could be included in the model, but the high multicollinearity between those variables indicated the need to drop them from the final model. Therefore, it was decided to discard those variables, obtaining a simpler model to predict BA for unthinned longleaf pine stands with similar goodnessoffit which avoided overfitting problems. Widely used models for other southern pines [
In the case of thinned stands, the use of the approach proposed by Pienaar [
The models that predict VOL (outside and inside bark) did not depend on stand age or N, and were similar or slightly better than other models reported for longleaf plantations [
One of the most important contributions of the system of equations presented in this study, in contrast to other longleaf models published, is the inclusion of the merchantable volume yield breakdown model. Similar to
The overall evaluation, where N, H_{dom}, BA and VOL_{OB}, were calculated for all unthinned plots using the system of equations shown in
Despite the fact that the model system performed very well for the dataset used for validation, the functioning of the model outside the geographical range of the fitting data is uncertain. We strongly recommend using this system of equations only within the range of data used to fit (see
This research was supported by the U.S. Department of Defense, through the Strategic Environmental Research and Development Program (SERDP). The authors acknowledge the U.S. Forest Service Southern Research Station for their assistance and for providing the long term datasets.
The authors declare no conflict of interest.