The authors have declared that no competing interests exist.

Accurate predictions for wood products classified by merchantable size are a matter of interest for forest managers and forestry companies, in order to estimate the monetary value of some of the many commodities and services that forests provide to society. The accuracy of this estimate is directly related to its final use. Thus, while forest managers need it for planning and quantification of land use, forestry companies need it to assess the profitability of a harvest.

Nowadays, society demands multifunctionality from our forests, which increases the need for more detailed knowledge of the different products and services a forest can provide. Product classification thus becomes an important tool for assessing the role of forest and wood products, especially in light of climate changes due to carbon fixation and sequestration. Volume predictions by different merchantable sizes facilitates further assessment of the life cycle of wood products by making it possible to precisely estimate the carbon stored in every wood product and subsequently evaluate entire forests (

Volume prediction to any merchantable limit can be achieved by several methods, most of which involve the use of volume-ratio or stem taper equations (

Compatibility means that an integrated model can be obtained through summation of the differential model. Thus, for a given merchantable volume equation, there is an intrinsically defined compatible taper function (

To develop a taper function, diameter/height data pairs are required along the stem. Most taper functions can be included in the following groups: single and segmented taper models, trigonometric equations, and variable-form taper models. Stem taper functions are usually based on the diameter at breast height (

Forestry researchers in Spain have been developing taper equations since the 1970s. Recently, a considerable number of taper equations have been developed for particular regions and species: some for hardwoods but most for softwoods (

In central Spain, most harvested tree species are Scots pine (^{3} and produce an annual harvest of about 1,274,594 m^{3}.

The objective of this study was to compare a stem taper function and a compatible merchantable volume system to ascertain which provides a better description of the stem profile, in order to obtain accurate partial or total stem volume estimates for the main species in the Spanish plateau (central Spain).

This research was carried out in the region of Castile-Leon, located in Central Spain. The region covers approximately 9.4 million ha and is one of the most important areas for timber production in Spain. Predominant oak stands cover more than half of the area, while pine stands cover one third of the area. Altitude fluctuates between 110 and 2650 m above sea level. The climate in Castile-Leon is both continental and Mediterranean: average winter temperatures range between 4 and 7 °C, while average summer temperatures range from 19 to 22 °C; with three or four dry summer months that are typical of a Mediterranean climate. Average annual rainfall is only 450 to 500 mm, mostly in lower altitudes.

The data used in this study were collected in 242 public and private forests in Central Spain. Data from 1,844 Scots pines, 456 stone pines, 533 black pines, 1,715 Mediterranean maritime pines, 326 Spanish junipers, 302 Pyrenean oaks, 992 poplars and 189 beeches were used to test the statistical performance of the taper functions. Thus, a total of 6,357 trees were selected for destructive sampling using the protocol of ^{3}) was obtained by summing the log volumes. Between 3 and 40 disks were cut per tree, for a total of 87,568 disks. Summary statistics including the number of observations, arithmetic mean, standard deviation, and minimum/maximum values of the main variables are presented in

Numerous taper functions have been developed and many describe the diameter along the stem quite well. Among them, the segmented function of

Variable-exponent taper equations describe the stem shape with a changing exponent from ground to top. This approach is based on the assumption that the stem form varies continuously along the length of a tree. The Stud model is basically an allometric function of the form ^{q}, where u is an exponential function that describes the butt region and _{1} and θ_{2} describe the upper and middle stem, respectively. The other parameters pertain to the function _{3} refers to width, θ_{4} refers to length and θ_{5} to height. The expression of this model is:

The Fang system assumes three sections with a variable-form constant factor for each one. The expression of this model is as follows:

where:

and a_{i}, b_{i} and p_{i} are the parameters to be estimated. ^{3}) and total volume (V, in m^{3}) by direct integration of the taper model. Their expressions are:

Although

The Stud model has been widely used with excellent results in radiata pine (

The models were fitted using least squares techniques. However, there are several problems associated with stem taper function analysis that violate the fundamental least squares assumptions of independence of errors: the two most important are multicollinearity and autocorrelation (

We followed the indications of

Among the different options to estimate the parameters in the Fang systems, where the taper equation includes a total volume equation, in this study we prioritized the taper function, fitting it first and subsequently performing the predicted volume calculation from the estimation parameters obtained (

Estimates of the different fitted models were compared by numerical and graphical analyses. Four goodness-of-fit statistics were used: the adjusted coefficient of determination (R^{2}
_{adj}), the mean bias error (BE), the root mean square error (RMSE), and the Bayesian Information Criterion (BIC).

The taper functions were also assessed using box plots for diameter residuals (

An n-way cross-validation of each species was carried out, estimating the residual for one tree by excluding that tree every time. The RMSE and the adjusted model efficiency (MEF_{adj}), equivalent to the R^{2}
_{adj} of the fitting phase) were calculated from the residuals.

To evaluate whether the taper equations vary among the different species, the nonlinear extra sum of squares method was used (

where SSE_{R} is the error sum of squares of the reduced model, SSE_{F} is the error sum of squares of the full model, and df_{R} and df_{F} are the degrees of freedom of the reduced and full models, respectively.

The models were first fitted without expanding the error terms to account for autocorrelation, thus a strong autocorrelation among all models was observed. A first-order continuous autoregressive error structure was required to model the inherent autocorrelation of the hierarchical data. An exception was found in the case of Spanish juniper in the Fang system, where the model did not converge. The autocorrelation may be explained by the effect of stand conditions (e.g., stand density, thinning effects, etc.) on stem form (

All the parameters were significant at p<0.05 (_{11} and a_{4} were not significant. The detailed error analysis of diameter predictions showed a similar trend in all the fits evaluated (sixteen combinations of the residuals, two models for each species). Although the poplar analysis rendered poorer graphics than the other species, all cases indicated a random pattern of residuals around zero (_{STUD} = 2.7241; RMSE_{FANG} = 2.6939), while the largest bias occurred in beech (BE_{STUD }= –0.1057; BE_{FANG }= –0.1882). By contrast, smaller RMSE were found in poplar plantations (RMSE_{STUD} = 0.7531; RMSE_{FANG} = 0.7574) and smaller BE in Scots pine (BE_{STUD }= -0.0689; BE_{FANG }= -0.0314). In all species, the Stud model and Fang system showed a similar behavior pattern for the root mean square error against classes of diameter at breast height: accuracy decreased with increasing size class (

The box plots of diameter residuals against relative height classes (

Results of the fitting process for full and reduced forms of the model are shown in _{58-71}=505.73) and the smallest differences occurred between black pine and Mediterranean maritime pine (F_{25-26}=4.63).

Numerous taper equations have been developed for all but two of the species analyzed. Stem taper functions for Spanish juniper and Pyrenean oak have not yet been developed. This paper provides a good starting point for fitting these species, for which only growth (

The results for Scots pine were similar to other findings (

The non-linear extra sum of squares method indicated that the stem taper differs among the five softwood species and three hardwood species. All of the 10 possible paired comparisons in softwoods and the 3 possible paired comparisons in hardwoods produced significant F-values. In softwoods, the greatest differences (as inferred from the F-values) occurred between black pine and Spanish juniper (F-value = 477.94) while the smallest differences were found between black pine and Mediterranean maritime pine (F-value = 4.63). In hardwoods, the greatest differences appeared between beech and hybrid poplar (F-value = 505.73) while the smallest differences were observed between Pyrenean oak and beech (F-value=41.93). These results are probably due to the strong apical dominance of the hybrid poplar compared to the Pyrenean oak and the beech, and their systematic sylviculture. All the pines studied were found to have the first inflection point at around 10% of total height, and the second inflection point at around 70% of total height (

We are grateful to the Forest Services (