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 [9]. The data were collected from regularly remeasured plots in a combination of seven studies exploring the effects of spacing and thinning on longleaf plantations distributed through the Western Gulf Coastal Plain from Santa Rosa County in Florida to Jasper County in Texas (Figure 1) and representing its current range in the Western Gulf Coastal Plain [9,10]. Plantations were established on both old field and cutover sites. Soil texture for plots were primarily silt loams, very fine sandy loams, or fine sandy loams, characteristic of the U.S. Upper Coastal Plain. Most plots were burned regularly by prescribed burns or wild fires. Each plot was measured for ~40 years at ~five-year intervals, averaging eight measurements per plot. Plots were rectangular and ranged in size from 0.04 to 0.1 ha⁻¹ [5].

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 [11] that relates H to the inverse of dbh, individual plot by measurement time regressions were determined (P < 0.0001) to estimate H in all trees without H measurement (MAE% = 4.3%; RMSE% = 5.6%; Bias% = −0.4%; R² = 0.97). Mean dominant height (H_dom, m) was determined for each plot at every measurement time as the mean of the top 25th percentile tree height. Site index (SI, m) was defined as H_dom at a reference age of 50 years after planting. As several plots were not measured at exactly age 50 yrs, SI was assessed using H_dom at index age plus/minus one year if necessary (i.e., 49 and 51 years).

From the complete dataset, 30 plots were randomly selected and removed to use for model evaluation and the rest (i.e., 237 plots) for model fitting. A total of 81 plots were thinned to constant basal area levels at five-year intervals; however, only pre-thinning measurements were considered on those plots. Summary statistics of individual trees and stand characteristics of both sub-datasets are shown in Table 1.

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 Table 2.

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.

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 d and t were selected to evaluate this model. The selected values of d and t correspond to the threshold values used to estimate breakdown volume yield for other southern pine species [25,26,27], corresponding to chip-and-saw (d = 21.6 cm; t = 10.2 cm) and sawtimber (d = 29.2 cm; t = 20.3 cm) products.

Four measures of accuracy were used to evaluate the goodness-of-fit 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 (R²) [28,29,30,31]. For BA and VOL, the statistics MAE, RMSE and Bias were back-transformed from logarithmic values.

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 [5], Brooks and Jack [7] and Lauer and Kush [32], (ii) dominant height equations reported by Farrar [4] and Brooks and Jack [7], and (iii) VOL_OB equations reported by Lohrey and Bailey [5] and Brooks and Jack [7]. The breakdown volume outside bark yield function was also compared against the functions reported for Pinus taeda [25] and Pinus elliottii [26] across different Dq and stand densities, and for three product classes (sawtimber: d = 29.2 cm and t = 20.3 cm; chip-and-saw: d = 21.6 cm and t = 11.2 cm; pulpwood: d = 11.4 cm and t = 5.1 cm), assuming stands with VOL_OB of 100 m³·ha⁻¹. This value of VOL_OB was selected to facilitate percent comparisons. The results of the breakdown volume yield function are independent of the value assumed.

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) [33]. When multiple linear regressions were carried out, the variance inflation factor (VIF) was monitored to detect multicollinearity between predicting variables, discarding all variables included in the model with VIF larger than 5, as suggested by Neter et al. [34]. In the case of BA and VOL fitting, where multiple linear regressions were carried out, step-wise procedure was used with a threshold significance value of 0.15 as variable selection criteria to enter and to stay [34]. For these responses, a logarithm transformation was preferred as it allows controlling for heterogeneity of variances, approximate to normality and uses the linear model framework to select among the large set of, potentially collinear, predicting variables.

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 Table 3. All parameter estimates were significant at P < 0.05. Non-linear versions of the models presented in Equation 6 (BA) and 8 (VOL) were also evaluated, but these resulted in no improvement in model performance (data not shown), therefore, natural logarithm-transformed response variables were used. Parameter estimates for the intercept in Equations 6 (c₁) and 8 (d₁) include the correction proposed by Snowdon [35]. The correction factor proposed by Baskerville [36] was also evaluated, but it presented lower bias reduction (data not shown).

Parameter estimates for the model that projects dominant height (Equation 2) are shown in Table 3. For SI of 20 m, the model projects dominant height of 5.4 and 22.6 m at age 10 and 70 yrs., respectively. If SI increased to 29 m, the model projects dominant height of 7.9 and 32.8 m at age 10 and 70 yrs., respectively. For all 194 plots where SI was measured, the mean observed and predicted SI was 25.84 and 25.88 m, respectively.

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 (i.e., between 20 and 29 m, see Table 1 and Table 2) and using a planting density of 1500 trees ha⁻¹ showed little mortality and only small differences in survival at age 10 yrs. (between 1450 and 1429 trees ha⁻¹, for SI 20 and 29 m, respectively). At age 70 yrs., however, the model estimated large differences in survival across SI’s (between 493 and 300 trees ha⁻¹, for SI = 20 and 29 m, respectively).

After applying the step-wise procedure and checking variance inflation factors (VIF), the final selected model that predicts BA (Equation 6) was only dependent on N and H_dom (Table 3). Although the variables 1/Age, ln(N)/Age and ln(H_dom)/Age were significant after the step-wise variable selection procedure (P < 0.001, data not shown), their VIF’s were high with values of 296, 173 and 35, respectively (data not shown). Therefore, these variables were discarded from the model and the goodness-of-fit of the final model was lower than the full model, having a CV of 3.6% and a R² of 0.944. Partial R² of H_dom and N were 0.579 and 0.366, respectively (data not shown).

The final model that predicts stand volume (Equation 8), after the step-wise variable selection procedure and VIF checking, was dependent on N, BA, ln(BA)/Age and SI (Table 2). The variable H_dom was discarded from the final model, even though they were selected after step-wise variable selection procedure (P < 0.001, data not shown), due to its high multicollinearity (VIF = 42.3, data not shown). The final models that predict stand V_OB and V_IB had a CV of about 1% and R² greater than 0.99. Stand BA explained most of the variability in V_OB and V_IB, with partial R² of about 0.912 and 0.867, respectively. Stand density presented partial R² of about 0.079 and 0.121, for V_OB and V_IB, respectively. Even though SI and ln(BA)/Age were significant, both explained less than 0.1% of changes in stand volume (data not shown).

The model that projects the time trend of CI after thinning (Equation 10) was dependent on stand age (Table 3). The exponential of the coefficient for a 36 year-old stand (mean values of stand age reported in Table 1), represents an average annual decline rate of the CI as the stand ages after thinning of 4.3%.

Parameter estimates for merchantable volume yield breakdown function (Equation 12) for both, outside (VOL_m-OB, m³·ha⁻¹) and inside (VOL_m-IB, m³·ha⁻¹) bark total volume yield, are shown in Table 3. The models had a CV of about 11% and an approximate R² of 0.99 for both outside and inside bark volume yield breakdown estimates.

There was a good agreement between predicted and observed values of N (Figure 2a), H_dom (Figure 2c) and BA (Figure 2e). The slope and the intercept of the relationship between predicted and observed values were not statistically different from one (P = 0.32) and zero (P = 0.14), respectively.

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 (Figure 2b for N; Figure 2d for H_dom and Figure 2e for BA) or stand age (data not shown).

A growth model to project, or update, BA (BA_j) when some stand measurements are available, including current BA (BA_i), along with current (i) and future (j) Age, N and H_dom, was derived from the fitted BA model (Equation 5), and is expressed as: (13)

As expected, this projecting growth model improved the estimations of BA, reducing the residuals as compared with BA predicting model (Figure 2g,h).

There was good agreement between predicted and observed values for VOL outside and inside bark (Figure 3a,c). In both cases, the intercept of those relationships was not statistically different from zero (P > 0.16). The slopes of the relationship between predicted and observed values (1.006 and 1.011, for VOL_OB and VOL_IB, respectively) were statistically different to one (P < 0.02). There was a good agreement between predicted and observed values of BA_t (Figure 3e). The intercept and slope were not different from zero (P = 0.18) and one (P = 0.12), respectively. If residuals are expressed as a percentage of observed value, maximum residuals observed in Figure 3 represent about 15% and 17% of observed VOL_OB and VOL_IB, respectively. There was no noticeable trend in residuals with observed values (Figure 3b for VOL_OB; Figure 3d for VOL_IB and Figure 3e for BA_t) or stand age (data not shown).

For the two combinations of t and d tested, there was good agreement between predicted and observed VOL_m (Figure 3g,i). For the two examples shown in Figure 3, the slope and intercept of those relationships were not different from one (P > 0.41) and zero (P > 0.36), respectively. For all variables listed above, there was no noticeable trend of residuals with observed values. Only for sawtimber VOL_m larger than about 500 m³·ha⁻¹ (Figure 3i,j), there was a small tendency to increase residuals as VOL_m increased, but the magnitude of that overestimation was less than 5% of observed values.

All model performance tests showed that N, H_dom, BA, BA_t, VOL and VOL_m estimations agreed well with measured values (Table 4). For all estimations, MAE and RMSE ranged between 2.4% to 10.3%, and 3.4% to 13.4% of the observed values, respectively. In all cases, BA estimations presented the larger differences between the observed and predicted values. The Bias ranged between 1.9% under-estimations for projected BA and 1.4% over-estimations for BA_t, with no clear tendency to over- or under-estimate. Estimated and observed values were highly correlated, with R² values greater than 0.91. The performance of the BA model that included the variables with high collinearity was also tested, showing lower MAE and RMSE than the final model (8.8, 12.6%, respectively) and larger absolute Bias (−3.3%) (data not shown).

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 (Figure 4). Across three stand age classes (<25, 25–49 and 50–75 yrs.), the models predict stand growth consistently, with no clear trend to over- or under-estimate. For example, N and H_dom estimations of all models performed adequately with Bias less than 10% (Figure 4a) and RMSE less than 15% with no apparent trend to change across stand age classes (Figure 4b). The estimates of VOL_OB were similar for all models for age less than 25 yrs., but for older stands the model reported by [7] Brooks and Jack (2006) over-estimated VOL_OB by around 70 m³·ha⁻¹, while the model reported by [5] Lohrey and Bailey (1977) over-estimated VOL_OB by about 50 and 10 m³·ha⁻¹ for stand age between 25–49 and 50–75 yrs., respectively.

Similarly, the model presented in this study under-predicted VOL_OB by about 30 m³·ha⁻¹ (or around 7%) for older stands (Figure 4e). The RMSE of VOL_OB estimations for the models reported in literature increased with stand age, averaging about 85 m³·ha⁻¹ at age class 25–75 yrs., whereas the model presented in this study had an error of about 19 m³ ha⁻¹ at age class 25–49 yrs., and 42 m³·ha⁻¹ at age class 50–75 yrs. (Figure 4f).

Examples of merchantable yield breakdown function of VOL_OB estimations for P. taeda [25], P. elliottii [26] and P. palustris (this study) are presented in Figure 5, showing, notably, similar results across species.

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 P. taeda, P. elliottii and P. palustris, respectively (Figure 5a,b). In the case of chip-and-saw yield, when Dq was 10 cm, all models predicted no volume production for that product, which has a threshold dbh limit of 21.6 cm. Independent of N, when Dq was larger than 50 cm, sawtimber yield was larger than 95% of VOL_OB and the production of chip-and-saw (Figure 5c,d) and pulpwood (Figure 5e,f) declined when the stands reached Dq larger than the merchantability limit for sawtimber. Overall, the merchantable yield breakdown functions presented in this study showed the expected behavior of product partitioning as Dq and N changed.

The overall test of the model indicated that, if only initial (i.e., current) stand age, N and SI (reported at first measurement) are known, estimations of H_dom, BA and VOL_OB were not affected, in relative terms, by simulation length (Figure 6d,f,h). On the other hand, projections of N were sensitive to the length of the simulation (Figure 6b), with errors getting larger as simulation length increased. In all cases residuals were centered on zero. For all stand parameters simulated, there was no noticeable trend of residuals with observed values, and the slope and intercept of the relationships between observed and predicted values were not different from one (P > 0.24) and zero (P > 0.42), respectively.

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% (Table 5). Across simulation lengths, the overall bias of the model system for N, H_dom, BA and VOL_OB were over-estimations of about 6 trees ha⁻¹ and under-estimations of about 0.2 m, 0.9 m²·ha⁻¹ and 13.4 m³·ha⁻¹, respectively. The overall MAE and RMSE of the model system were about 12 and 18% for N, 3 and 3% for H_dom, 12 and 16% for BA and 13 and 18% for VOL_OB, respectively. The R² decreased as simulation length increased. The overall R² across simulation lengths were about 0.93, 0.97, 0.84 and 0.84, for N, H_dom, BA and VOL_OB, respectively. A trend of increasing error with simulation length was observed for Bias, MAE and RMSE (Table 5). Nevertheless, it is important to note that the number of observations decreases as the simulation length gets larger (Table 5), and therefore, the evaluation statistics on simulation lengths ~40 yrs. can be affected by the unbalanced sampling size and specific characteristics of the sampled plots for evaluations.

An example of model behavior for a hypothetical longleaf stands planted with 1400 trees ha⁻¹ is shown in Figure 7. The unthinned stands growing on a site with SI = 20 m reached at age 70 yrs. a survival of about 46% of initial planting density, H_dom of 22.6 m, BA of 28.5 m²·ha⁻¹, SDI of 571 tress ha⁻¹ and VOL_OB of 338 m³·ha⁻¹. When SI was 30 m instead, at the same age the survival was 31% and H_dom was 33.8 m. The unthinned stand reached a maximum BA of about 48.1 m²·ha⁻¹ at age 58 yrs., and SDI peaked at about 870 trees ha⁻¹ at age 49 yrs. and VOL_OB was still increasing, reaching about 610 m³·ha⁻¹ at age 70 yrs. When a scenario of three thinnings was applied to both stands, at age 70 yrs. the number of surviving trees was about 181 and 122 trees ha⁻¹ and Dq was increased from 24.6 and 38.3 cm (unthinned), to 30.2 and 47.2 cm, for SI of 20 and 30 m, respectively. The harvested volume from thinnings was about 171 and 331 m³·ha⁻¹, and the final yield was 162 and 293 m³·ha⁻¹ for SI of 20 and 30 m, respectively.