 Research article
 Open Access
Use of upper stem diameters in a polynomial taper equation for New Zealand radiata pine: an evaluation
 Charles O. Sabatia^{1, 2}Email author
https://doi.org/10.1186/s4049001600702
© The Author(s). 2016
 Received: 20 December 2015
 Accepted: 15 July 2016
 Published: 20 August 2016
Abstract
Background
Polynomial taper models are the tree profile equation types that are most commonly used to describe stem profiles of New Zealand radiata pine (Pinus radiata D.Don). Among these equations are model forms that include an upper stem diameter measurement as a predictor. Such taper functions may be more costly to use given the need to measure the additional stem diameter. Therefore, it is important to have an insightful understanding of the contribution of the additional diameter measurement to taper model prediction bias and precision to better inform decisions on whether to develop advanced technology for, or invest in, tree upper stem diameter measurements(s) for taper equation use.
Methods
Prediction precision and bias for diameter and volume were evaluated for a regular polynomial taper model with dbh and total height as treelevel predictors (DH model) and eight taper models that included an upper stem diameter as an additional treelevel predictor (DH+ models). Two sets of radiata pine stem sectional data from 66 stands across New Zealand were used in the investigation. Seventy percent of the trees in the larger one of the two sets were used for model fitting. The rest of the trees in the larger data set, and those in the remaining data set, were used for model validation.
Results
A DH+ taper equation that included an upper stem diameter measured at half the distance between breast height and the tree tip exhibited the lowest prediction error for stem diameters. Most of the reductions in diameter prediction error, from use of this equation, occurred in the upper half of the tree bole. Treelevel volume predictions from both DH and DH+ taper models were unbiased across the range of tree sizes investigated, but DH+ models exhibited greater precision in volume prediction.
Conclusions
Including an outside bark upper stem diameter, measured at half the distance between breast height and the tree tip, in a polynomial taper equation for New Zealand radiata pine may result in a considerable improvement in tree volume prediction precision especially for total aboveground volume. Use of a diameter measured at 6 m may not result in prediction precision that is different from that of a DH taper equation.
Keywords
 Pinus radiata
 Stem taper
 Tree profile equations
 Tree volume estimation
Background
Polynomial stem taper equations are the type of tree profile equation that are most commonly used to compute stem volume estimates that support forest management decisions in New Zealand. The most widely used version of these equations is the compatible polynomial taper equation (Goulding and Murray 1976; Gordon 1983; Katz et al. 1984; Hayward 1987). A compatible polynomial taper equation is constructed to include algebraic constraints that ensure that the estimate of total stem volume from the analytical integration of the taper equation is equal to the estimate from an associated tree volume equation (DiéguezAranda et al. 2006). Regular polynomial taper models, without volume compatibility constraints, have also been fitted for some of the tree species grown in New Zealand (e.g. Gordon et al. 1995, 1999). Such taper models are not constrained to guarantee volume predictions that equal those from the associated tree volume equation and hence are free from possible effects of volume compatibility constraints on taper equation predictive ability.
Including upper stem diameter(s) as additional taper model predictor(s) is a widely used method of accounting for intertree stem profile differences in stem taper functions. Various approaches of incorporating the diameter(s) in taper models have been proposed and demonstrated, which include use of upper stem diameters in a taper model equation system (e.g. Kilkki et al. 1978; Kilkki and Varmola 1979), use of the diameter(s) as direct predictors or as algebraic constraints (e.g. Czaplewski and McClure 1988; Kozak 1998; Cao 2009; Sabatia and Burkhart 2015) or use of the diameter(s) to compute the empirical Bayes estimate of the random effects taper equation parameter(s) of the subject tree (e.g. Sharma and Parton 2009; Cao and Wang 2011; AriasRodil et al. 2015; Sabatia and Burkhart 2015). Results from these studies suggest that the best upper stem diameter to include in a taper model probably differs by species, by taper model form and/or by region of the world. Most published studies have pointed to an upper stem diameter measured at approximately 50 % of total height, but others have pointed to diameters measured at as low as 38 % of total height and others at as high as 60 % of total height.
In taper models for New Zealand radiata pine (Pinus radiata D.Don), use of upper stem diameters was first demonstrated by Gordon and Budianto (1999) who developed a polynomial taper model that included an upper stem diameter, measured at 6 m above the ground (approximately 16 % of the average height of a mature New Zealand radiata pine tree), as an algebraic constraint predictor. The authors did not evaluate the role of the upper stem diameter in this equation nor did they evaluate other upper stem diameter measurements. Therefore, potential users of the ‘3point volume and taper equation for radiata pine’ of Gordon and Budianto (1999) do not have a basis of deciding whether use of this model, as opposed to a dbhandtotalheightonly model (hereinafter referred to as the DH model), is beneficial. In addition, it is important to identify the upper stem diameter that is best to use in this model type to provide informed recommendations for users and add to the knowledge on the question of whether or not the most effective upper stem diameter to include in a taper model may differ by model type. In a recent study, Sabatia and Burkhart (2015) investigated the use of upper stem diameters in a segmented Max and Burkhart (1976) taper equation for New Zealand radiata pine. This study used a model form that is not currently used in New Zealand forestry, and hence, the research related more to the general scientific interest of taper equation researchers than it did to New Zealand users of taper equations. The current study seeks to follow up on the findings of Sabatia and Burkhart (2015) using a taper model that is one of those currently used in New Zealand forestry.
The objectives of the current study were to identify the most effective upper stem diameter for inclusion in a regular, novolumecompatibilityconstraint, polynomial taper model for radiata pine in New Zealand and to determine the effect, on tree volume prediction precision and bias, of including the upper stem diameter as a predictor in the taper model.
Methods
Data
Measurements on each tree in the data sets included dbh (breast height = 1.4 m above the ground), total height, outside bark diameter and bark thickness measurements at select points along the tree bole. The points were 0.15, 0.7, 1.4 (breast height) and 3 m above the ground and thereafter every 3 m up to the point where the remaining distance to the tip was less than 3 m. Diameter measurements were taken on felled trees. Total aboveground outside bark volume was calculated for each tree by applying Smalian’s formula to each of the bolt sections between stump height and the last diameter measurement point up in the stem, then applying volume of a cylinder formula to the stump (because groundline diameters were not measured) and volume of a cone formula to the tip portion and, lastly, summing up the bolt volumes. Merchantable volume to 14cm outside bark top diameter (pulpwood top diameter limit) and to 22cm outside bark top diameter (sawtimber top diameter limit) were calculated for trees that had an abovestump merchantable height of at least 3.8 m by applying Smalian’s formula to each of the bolt sections between stump height and the top diameter limit and then summing up the bolt volumes. Where height at 14 or at 22cm top diameter was not actually measured during data collection, it was computed by quadratic interpolation using measured heights at three closest diameters whose range included 14 or 22 cm, whichever was applicable. The specified diameter limits and the merchantable height requirement of at least 3.8 m were based on the current New Zealand domestic and export markets small end diameter and log length limits for radiata pine (Laurie Forestry 2016).
Summary statistics of the dbh and tree total height of the trees in the training, splitvalidation (Splitval) and independentvalidation (Indval) data
Statistic  dbh (cm)  Total height (m)  

Training data  Splitval data  Indval data  Training data  Splitval data  Indval data  
Minimum  15.9  19.9  36.6  19.4  15.4  25.6 
Median  47.0  46.2  54.3  35.3  35.1  37.1 
Maximum  76.1  76.8  77.8  49.9  49.7  48.3 
The taper models
In Eqs. 2 to 6, \( {z}_{bh}=\frac{H1.4}{H} \) and \( {z}_6=\frac{H6}{H}, \) D _{6} is the measured diameter at 6m height, and H is the total height. The rest of the terms are as defined for Eq. 1. According to the authors of the GB99 model, the roles of Eqs. 4, 5 and 6 in the taper model are to make it respond to changes in tree size. The GB99 model was one of the DH+ models evaluated in the current study.
Model M3 was a more parsimonious generalized version of GB99. Upper stem diameter measurements (D _{ us }) from seven different points along the stem (h _{ DUS }) were used in M3 resulting in seven different versions of this model, which were evaluated to identify the most effective D _{ us } to use in this model form. The seven h _{ DUS } points were (1) a point at 50 % of the height above dbh and (2) a point at 20 % of the total height of the tree and then a point every 10 percentage points up to 70 % of the total height of the tree. When h _{ DUS } occurred where diameter was not actually measured during data collection, D _{ us } for the h _{ DUS } was computed by quadratic interpolation involving a closest measured diameter below and a closest measured diameter above the h _{ DUS } plus one other measured diameter that was next closest to the h _{ DUS }.
Model fitting

I _{1} = 1 for the bole part below breast height and 0 otherwise

I _{2} = 1 for the bole part between dbh and the upper stem diameter measurement point and 0 otherwise

I _{3} = 1 for the bole part above the upper stem diameter measurement point and 0 otherwise

I _{1} = 1 for the bole part below breast height and 0 otherwise

I _{2} = 1 for the bole part above dbh and 0 otherwise
Equations 11 and 12 are based on the reasoning that error variance will depend on tree size (represented by dbh) and distance from the ground or distance from the constraint point. Constraint points (breast height and height at the upper stem diameter measurement point) were considered as the other starting points, aside from the ground level, because error variance at these points is 0 (because the models investigated are conditioned to predict the observed dbh and the observed upper stem diameter).
Model evaluation
The fitted models were evaluated for fit on the fitting data set and for prediction performance on the splitvalidation and independentvalidation data sets. All models were evaluated for diameter fit and prediction performance, but only the BASE, GB99 and the best one of the seven M3 models were evaluated for volume prediction performance. Modelpredicted volume was calculated from the modelpredicted diameters using Smalian’s formula as was done for the observed tree volumes (see the ‘Data’ section of the current paper for the description of the observed tree volume calculations).
Diameter and volume prediction performance ranking of the models was based on the MAB and RMSE statistics. The MB statistic was not used in the ranking process because it does not capture large positive and negative deviations, which cancel out when deviations get summed during MB calculation.
Results
Fit statistics for diameter prediction by the seven versions of M3 taper models fitted on the ‘fitting’ data set
h _{ DUS }  Fit statistic  

MB  MAB (%)  RMSE  
20 %  −0.331  3.758  5.404 
30 %  −0.249  3.434  4.995 
40 %  −0.077  3.243  4.731 
50 %  0.208  3.108  4.543 
50 % HABH  0.263  3.093  4.523 
60 %  1.201  3.404  4.982 
70 %  0.583  3.406  4.978 
Parameter estimates for the BASE, GB99 and M3_HABH taper models
Parameter  Estimate  

BASE model  Model GB99  Model M3_HABH  
β _{1}  TS  TS  TS 
β _{2}  –  TS  TS 
β _{21}  0.0929 (0.0130)  –  – 
β _{3}  0.4125 (0.0189)  –  0.5502 (0.0053) 
β _{30}  –  0.8213 (0.0351)  – 
β _{31}  –  −0.2040 (0.0199)  – 
γ _{1}  1.4420 (0.0061)  –  1.7149 (0.0440) 
γ _{10}  –  0.9377 (0.0237)  – 
γ _{11}  –  0.3522 (0.0169)  – 
γ _{2}  16.6627 (2.4157)  14.5371 (0.6206)  2.1955 (0.1190) 
γ _{3}  64.2933 (3.0251)  –  34.3438 (0.4982) 
γ _{31}  –  1.3431 (0.0351)  – 
Diameter fit/prediction and volume prediction error statistics for the BASE, GB99 and M3_HABH taper models
Dimension  Model  Fitting data  Splitvalidation data  Independentvalidation data  

MB_{dob} (%)  MAB_{dob} (%)  RMSE_{dob} (%)  MB_{dob/v } (%)  MAB_{dob/v } (%)  RMSE_{dob/v } (%)  MB_{dob/v } (%)  MAB_{dob/v } (%)  RMSE_{dob/v } (%)  
Stem diameters  BASE  0.143  4.254  5.802  0.696  4.313  5.788  1.603  5.060  7.052 
GB99  −0.429  3.561  5.208  −0.174  3.529  5.110  1.005  4.291  6.136  
M3_HABH  0.263  3.093  4.523  0.466  3.142  4.488  0.535  3.872  5.615  
Total aboveground volume  BASE  1.133  5.515  7.720  1.371  5.322  7.408  
GB99  −1.316  3.768  5.247  0.525  4.683  6.372  
M3_HABH  0.768  2.729  3.824  −0.816  2.810  3.937  
Volume to 14cm top diameter (pulpwood volume)  BASE  0.784  5.328  7.553  0.707  5.092  6.989  
GB99  −1.586  3.671  5.153  −0.003  4.430  6.056  
M3_HABH  0.616  2.675  3.709  −1.232  2.863  4.140  
Volume to 22cm top diameter (sawtimber volume)  BASE  0.884  5.123  7.402  0.129  4.718  6.514  
GB99  −1.616  3.378  4.974  −0.472  4.038  5.697  
M3_HABH  0.763  2.861  3.982  −1.592  2.994  4.457 
The prediction error statistics for individual tree volumes are given in Table 4. The greater diameter prediction precisions of the GB99 and M3_HABH models relative to the BASE model, as expected, translated into greater precision for stem volume prediction. The improvement in volume prediction precision, due to inclusion of upper stem diameters in the taper model, was generally larger compared to the corresponding improvement in diameter prediction precision. Considering the better performing upper stem diameter model M3_HABH, gains in stem volume prediction precision were highest for total aboveground volume and lowest for sawtimber volume.
Discussion
For the polynomial taper model investigated in the current study, the diameter measurement at half the distance between breast height and the tip of the tree was the best diameter to include in a DH+ model. This finding is different from the one by Sabatia and Burkhart (2015) who, with the same data used in the current study, concluded that the diameter measurement at 60 % of total height was the best to include in a Max and Burkhart (1976) segmented taper model that had been constrained to dbh and an upper stem diameter. Nonetheless, the finding in the current study concurs with that by Cao (2009) who concluded that a diameter at half the distance between breast height and the tip of the tree was the best to include in a modified version of the Max and Burkhart (1976) segmented taper model that had also constrained to dbh and an upper stem diameter. In general, the finding in the current study points to a region in the neighbourhood of 50 % of total height as being the best point to measure an upper stem diameter to be included in a DH+ taper model, similar to what some previous studies have found (e.g. Kozak 1998; AriasRodil et al. 2015). The differences in findings between studies may be due to differences in taper model form.
With and withoutupperstemdiameter diameter prediction RMSEs for taper models investigated in previous studies
Author and year  RMSE units  RMSE  

Model without upper stem diameter  Model with upper stem diameter  
Kozak (1998)^{a}  %  12.56  9.91 
Kozak (1998)^{a}  %  10.80  8.45 
Sharma and Parton (2009)^{a}  \( \frac{\mathrm{dob}}{\mathrm{dbh}} \)  0.787  0.5909 
Sharma and Parton (2009)^{a}  \( \frac{\mathrm{dob}}{\mathrm{dbh}} \)  0.4915  0.3764 
Yang et al. (2009)  cm  0.7681  0.6708 
GómezGarcía et al. (2013)  cm  1.55  1.43 
Similar to the results with the segmented taper model in Sabatia and Burkhart (2015), use of the best upper stem diameters in the polynomial model in the current study mainly improved the performance of the taper model in the upper half of the tree bole (Figs. 3 and 4). Consequently, volumes of tree boles that are longer than 50 % of the tree’s total height will be predicted more precisely by a DH+ taper model than they would be predicted by a DH model. On the other hand, tree boles that do not include, or only include a small part of, the stem portion above 50 % of total height are unlikely to be predicted differently by a DH+ taper model compared to how they would be predicted by a DH one. This explains the observation that the decrease in volume prediction error, due to use of the M3_HABH taper model, was largest for total aboveground volume and smallest for sawtimber volume (Table 4). Sawtimber top diameter limits were, on average, at a relative height of 60 % for trees in the splitvalidation data set and at a relative height of 67 % for those in the independentvalidation data set. Pulpwood top diameter limits were, on average, at a relative height of 78 and 81 % for trees in the splitvalidation and for those in the independentvalidation data sets, respectively.
For prediction of diameters up to 60 % of total height, the BASE model performed better on trees in the independentvalidation data set than it did on those in the splitvalidation data set (Fig. 4a versus Fig. 4b). This observation was contrary to expectation as a model would be expected to perform better on data that is more closely related to the fitting data set. The possible explanation for this is that trees in the independentvalidation data set had profiles that were more clustered around the mean profile of the fitting data set where the BASE model, which was essentially a global model, is expected to perform better (see Fig. 2). Profiles of trees in the independentvalidation are likely to exhibit this pattern because the trees were from stands that were less dispersed geographically (Fig. 1), covered a narrower range of ages (23 to 25 years compared to 9 to 39 years for trees in the fitting and splitvalidation data sets) and covered a narrow range of stand densities (230 to 280 trees ha^{−1} compared to 150 to 700 trees ha^{−1} in the fitting and splitvalidation data sets). The difference in BASE model performance for the lower two thirds of the tree bole, between the splitvalidation and the independentvalidation data sets, can also be seen in the sawtimber RMSEs and magnitude of change in these RMSEs when the M3_HABH taper model is used instead of the BASE model (Table 4). Splitvalidation data sawtimber RMSE was larger, and it reduced by a greater magnitude (approximately 3.5 percentage points compared to close to 2.0 percentage points for the independentvalidation data set) when the M3_HABH taper model was used. Whether or not a data set, on which one wishes to apply a taper model, falls in the portion of the model fitting data where a global model performs best is in most cases unknown to a user. Thus, use of upper stem diameters in polynomial taper models could make the models robust to effects of unknown differences between model fitting and independent model application data sets.
Conclusions
If use of an upper stem diameter in a novolumecompatibilityconstraint polynomial taper equation for New Zealand radiata pine is desired, a diameter measured at half the distance between breast height and the tip of the tree would be the most effective upper stem diameter to use. Consequently, the model of choice would be the version of the Gordon and Budianto (1999) model in which a tree’s taper curve is constrained to pass through this upper stem diameter, i.e. M3_HABH. The original Gordon and Budianto (1999) model, GB99, which included an upper stem diameter measured at 6 m above the ground among its predictor variables, may not give predictions that are different from those of a polynomial model of the same form but without upper stem diameters among the predictor variables.
With use of the version of the Gordon and Budianto (1999) model that includes a diameter at half the distance between breast height and the tip of the tree (M3_HABH), considerable gains in stem volume prediction precision may be realized especially for total aboveground volume. It should, however, be noted that in such an application, gains in diameter and/or volume prediction precision may be lower if the measurement precision for the upper stem diameters is low. Upper stem diameters used in the current study were measured without error or were estimated (through quadratic interpolation) with a relatively high precision. It should also be noted that the reported volume prediction errors are based on outside bark diameters. Outside bark to inside bark diameter ratios may be used to adjust predictions from the reported taper equations if inside bark volumes are desired.
The study also showed that taper equation form may affect the decision on which upper stem diameter is best to include in the equation as an additional predictor. The diameter at half the height between breast height and the tip of the tree found to be the best in the current study was slightly different from the 60 % of total height found in an earlier study with a segmented model.
Declarations
Acknowledgements
Stem sectional data used in the reported analyses was provided by New Zealand Forest Research Institute Limited (Scion). Financial support for data analysis was provided by the Growing Confidence in Forestry’s Future radiata pine productivity project that was jointly funded by the New Zealand Ministry of Business, Innovation and Employment and the New Zealand Forest Growers Levy Trust. Data analysis for the work reported here was completed when the author was a scientist at Scion in Rotorua, New Zealand.
Author’s contributions
The author conceived the study, carried out all the analyses and wrote the manuscript.
Author’s information
The author is currently an Assistant Professor of Forest Biometrics at Mississippi State University, Mississippi State, MS, USA. Previously, he was a Scientist for Forest Modelling and Biometrics at Scion in Rotorua, New Zealand.
Competing interests
The author declares that he has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 AriasRodil, M., DiéguezAranda, U., Puerta, F. R, LópezSánchez, C. A., Líbano, E. C., Obregón, A. C., et al. (2015). Modelling and localizing a stem taper function for Pinus radiata in Spain. Canadian Journal of Forest Research, 45(6), 647–658. doi:10.1139/cjfr20140276.View ArticleGoogle Scholar
 Cao, Q. V. (2009). Calibrating a segmented taper equation with two diameter measurements. Southern Journal of Applied Forestry, 33(2), 58–61.Google Scholar
 Cao, Q. V., & Wang, J. (2011). Calibrating fixed and mixedeffects taper equations. Forest Ecology and Management, 262(4), 671–673. doi:10.1016/j.foreco.2011.04.039.View ArticleGoogle Scholar
 Czaplewski, R. L., & McClure, J. P. (1988). Conditioning a segmented stem profile model for two diameter measurements. Forest Science, 34(2), 512–522.Google Scholar
 DiéguezAranda, U., CastedoDorado, F., ÁlvarezGonzález, J. G., & Rojo, A. (2006). Compatible taper function for Scots pine plantations in northwestern Spain. Canadian Journal of Forest Research, 36(5), 1190–1205. doi:10.1139/x06008.View ArticleGoogle Scholar
 GómezGarcía, E., CrecenteCampo, F., & DiéguezAranda, U. (2013). Selection of mixedeffects parameters in a variable–exponent taper equation for birch trees in northwestern Spain. Annals of Forest Science, 70(7), 707–715.View ArticleGoogle Scholar
 Gordon, A. D. (1983). Comparison of compatible polynomial taper equations. New Zealand Journal of Forestry Science, 13(2), 146–155.Google Scholar
 Gordon, A. D., & Budianto, M. (1999). A 3point stem volume and taper equation for radiata pine. Rotorua: Forest and Farm Plantation Management Cooperative. Report No. 66.Google Scholar
 Gordon, A. D., Lundgren, C., & Hay, E. (1995). Development of a composite taper equation to predict overand underbark diameter and volume of Eucalyptus saligna in New Zealand. New Zealand Journal of Forestry Science, 25(3), 318–327.Google Scholar
 Gordon, A. D., Lundgren, C., & Hay, E. (1999). Composite taper equations to predict overand underbark diameter and volume of Eucalyptus pilularis, E. globoidea, and E. muelleriana in New Zealand. New Zealand Journal of Forestry Science, 29(2), 311–317.Google Scholar
 Goulding, C., & Murray, J. (1976). Polynomial taper equations that are compatible with tree volume equations. New Zealand Journal of Forestry Science, 5(3), 313–322.Google Scholar
 Hayward, W. J. (1987). Volume and taper of Eucalyptus regnans grown in the central North Island of New Zealand. New Zealand Journal of Forestry Science, 17(1), 109–120.Google Scholar
 Katz, A., Dunningham, A. G., & Gordon, A. D. (1984). A Compatible volume and taper equation for New Zealand Pinus radiata D. Don grown under the direct sawlog regime. Rotorua: New Zealand Forest Service. FRI Bulletin No. 67.Google Scholar
 Kilkki, P., Saramäki, M., & Varmola, M. (1978). A simultaneous equation model to determine taper curve. Silva Fennica, 12(2), 120–125.View ArticleGoogle Scholar
 Kilkki, P., & Varmola, M. (1979). A nonlinear simultaneous equation model to determine taper curve. Silva Fennica, 13(4), 293–303.View ArticleGoogle Scholar
 Kozak, A. (1998). Effects of upper stem measurements on the predictive ability of a variableexponent taper equation. Canadian Journal of Forest Research, 28(7), 1078–1083.View ArticleGoogle Scholar
 Lappi, J. (2006). A multivariate, nonparametric stemcurve prediction method. Canadian Journal of Forest Research, 36(4), 1017–1027. doi:10.1139/x05305.View ArticleGoogle Scholar
 Laurie Forestry (2016). Marketing services. http://www.laurieforestry.co.nz/ForestryMarketing. Accessed 2 July 2016.
 Max, T. A., & Burkhart, H. E. (1976). Segmented polynomial regression applied to taper equations. Forest Science, 22(3), 283–289.Google Scholar
 Sabatia, C. O., & Burkhart, H. E. (2015). On the use of upper stem diameters to localize a segmented taper equation to new trees. Forest Science, 61(3), 411–423. doi:10.5849/forsci.14039.View ArticleGoogle Scholar
 SAS Institute Inc. (2002  2004). SAS 9.4 Help and Documentation. SAS Institute Inc.: Cary, NC.Google Scholar
 Sharma, M., & Parton, J. (2009). Modeling stand density effects on taper for jack pine and black spruce plantations using dimensional analysis. Forest Science, 55(3), 268–282.Google Scholar
 Yang, Y., Huang, S., Trincado, G., & Meng, S. X. (2009). Nonlinear mixedeffects modeling of variableexponent taper equations for lodgepole pine in Alberta, Canada. European Journal of Forest Research, 128(4), 415–429.View ArticleGoogle Scholar
 Zimmerman, D. L., & NúñezAntón, V. (2001). Parametric modelling of growth curve data: an overview. Test, 10(1), 1–73.View ArticleGoogle Scholar