A Preliminary System of Equations for Predicting Merchantable Whole-Tree Volume for the Decurrent Non-Native Quercus rubra L. Grown in Navarra (Northern Spain)

Abstract

Estimation of tree volume typically focuses on excurrent forms, with less attention given to decurrent forms. Species with a decurrent form, particularly hardwoods, lack a dominant stem and have large diameter branches that can be included in the merchantable wood volume. We developed a preliminary two-equation system comprising a taper equation and a merchantable whole-tree volume (stem and branches) equation for Quercus rubra L. growing in Navarra (Northern Spain). The equation system includes the diameter at breast height and total tree height as independent variables, along with merchantable height—the height up to which the stem maintains a well-defined excurrent form—as an additional variable. After estimating the stem volume, the branch volume is estimated by subtracting the stem volume from the merchantable whole-tree volume. A second order continuous autoregressive error structure was used to correct for autocorrelation between residuals from the fitted taper equation. The equations explained 90% of the observed variability in diameter and 86% of the observed variability in merchantable whole-tree volume. Both equations have been implemented in the Cubica Navarra 3.0 software for use as a system of equations. These equations are considered preliminary and will be refitted or validated as additional data becomes available from new locations.

1. Introduction

Estimation of tree wood volume is one of the most important tasks for forest managers. Tree volume is usually considered equal to stem volume, and taper equations are used when product demand is expressed in terms of tree dimensions for different industrial uses. Taper equations predict the variation in diameter along the stem, and they therefore characterize stem form [1,2]. Integration of the taper equation from the ground to any height provides an estimate of the wood volume up to that height. However, in contrast to the almost always excurrent form in softwoods, hardwoods often have a decurrent form [3,4]; and not all trees have a well-defined stem and they may be forked or lack apical dominance. In the development of taper equations, trees that differ from the idealized excurrent form are generally excluded [5]. However, some authors have incorporated different stem forms in stem taper models. For example, Adu-Bredu et al. [6] considered three general forms of Tectona grandis L.f. in West Africa, in order to take into account the effect of forks in the taper equation: zero-forked trees, one-forked trees, and two-forked trees. For forked trees, the height(s) of fork(s) must be measured by the forest manager. MacFarlane and Weiskittel [5] developed a generalized stem taper which can be calibrated by adding random-effects parameters depending on the species (22 species were considered, including Quercus rubra L.) and merchantable form types (10 types were considered, including the combination of the presence/absence of merchantable stem/branches with the destination of saw/pulp).

On the other hand, some branches may be thick enough to be merchantable (Figure 1) as a subproduct [5], and their volume should be known. However, most research is focused on modelling stem volume, and estimation of branch volume has received less attention [7,8,9]. Quantification of tree branches has typically considered branch weight, as quantifying branch volume is challenging due to the large numbers of pieces of different sizes and shapes [7].

The northern red oak (Quercus rubra) was introduced in northern Spain for productive purposes in the 20th century, and Navarra region currently has the largest stocks of this species [10]. Red oak is cultivated to obtain straight, thick and clean stems, but there are trees that differ from the idealized excurrent form. Therefore, the Regional Forest Service, in addition to diameter at breast height (D, measured at 1.3 m above the ground) and total tree height (H), usually also measures the merchantable height (h_m, Figure 1). Merchantable height is the subjective height up to which the stem has a form similar to excurrent form. When only taper equations are developed using D and H, the following are assumed: (i) excurrent form occurs and the stem is considered by the minimum top diameter fixed by the forest manager (d_min, Figure 1 right); and (ii) merchantable whole-tree volume (V_t) would be equal to stem volume, and V_t could therefore be underestimated. It has been found by the Regional Forest Service that V_t is generally underestimated in trees classified as “poorly formed” because they have a large branch volume. Stem volume equations have already been developed for different stem forms and the main forest species considered in the Spanish National Forest Inventory (IFN in its original acronym) [11]. However, the IFN approach does not classify stem products or consider the branch volume.

The overall objective of the present study was to develop a preliminary system of equations that allows estimation of the merchantable whole-tree volume and classification of wood products for Q. rubra growing in Navarra. The specific objectives were as follows: (i) to develop a taper equation that enables classification of stem products; (ii) to develop an equation that estimates the merchantable whole-tree volume (V_t), for both stem and branches; and (iii) to determine the compatibility of both equations and their applicability in routine forest inventories in Navarra.

2. Materials and Methods

2.1. Data

The study was conducted in the region of Navarra (North of Spain). A total of 30 oak trees were destructively sampled in 2001 in the municipality of Etxalar in order to develop a taper equation for the Navarra Government. In 2021, sampling of 34 trees was carried out in the municipality of Altsasu. For each tree, the diameter at breast height (D, in cm, 1.3 m above ground level) was measured twice at right angles to each other and then averaged. Subsequently, the trees were felled, resulting in stumps of average height of 0.10 m. The maximum length of the felled trees was measured to the nearest 0.01 m to determine the total tree height (H, m). The bole was subsequently sectioned into logs at 2 m intervals in the trees from Etxalar and 1 m in those from Altsasu, up to a minimum diameter of 7 cm in the trees which had a well-defined dominant stem or, otherwise, until the merchantable height (h_m) was reached. Two perpendicular diameters over bark were measured in each cross section (at height h, in m, from ground level), and the values were then averaged (d, cm). Moreover, in Altsasu, the merchantable branches also were measured at 1-m intervals. For the branches, the same minimum diameter of 7 cm was set, except where branch architecture indicated the need for a larger top diameter. The volume of each log (from the stem or branches) was calculated by Smalian’s formula. The nomenclature used in this study is summarized in

Table 1. Abbreviations used in the study. See Figure 1 for clarification.

Denotation	Description
D	diameter at breast height (cm), measured at 1.3 m above the ground
H	total tree height (m)
hm	merchantable height (m)
d	diameter (cm) at a given height h
h	height (m) above ground to diameter d
dmin	minimum top diameter (cm), in this study dmin = 7 cm
dt	total stem top diameter (cm)
dwt	whole-tree top diameter (branches included) (cm)
Vt	merchantable whole-tree volume (branches included) (m3)
Vs	stem volume (m3) for excurrent form until H
vs	stem volume (m3) until h
Vb	merchantable branch volume (m3)

.

2.2. Taper Equation

The scatter plot of relative diameter (d/D) against relative height (h/H) was visually examined for the 64 available trees to detect any possible anomalies in the data. Because inconsistent data were detected, a systematic method was taken to identify abnormal data points [12], enhancing process efficiency. This involved employing local quadratic nonparametric fitting (LOESS) with a smoothing factor of 0.3, determined through iterative fitting and visual assessment of the smoothed taper curves overlaid on the data. This approach detected 1.28% anomalies from the initial data points, and these outliers were removed before parameterizing the taper models. Some of the outliers could be due to errors in measuring the bole sections or mistakes in transcribing field notes. However, others might be the result of deformations caused by fire damage, large knots, and other types of physical damage, such as necrosis, growth deformities, and so on. For the data ultimately utilized, the summary statistics are shown in

Table 2. Descriptive statistics of the sample of trees used for taper equation development.

Variable	Mean	Minimum	Maximum	Std. Dev.
Sections	14.5	6.0	27.0	4.99
D	37.5	25.0	58.5	8.20
H	24.5	17.5	31.4	2.86
dt	11.0	7.0	34.0	6.29

Sections, number of sections per tree; D, diameter at breast height (cm); H, total tree height (m); d_t, total stem top diameter (cm).

, while Figure 2 illustrates the scatter plot of d/D against h/H.

The taper equations tested include the models developed by Bi [12], Max and Burkhart [13], Biging [14], Stud [15], Fang et al. [16], Sharma and Zhang [17], and Kozak [18]. Firstly, the models were fitted by Ordinary Nonlinear Least Squares (ONLS) using the Gauss–Newton iterative method [19]. The initial values used in the iteration process were obtained from previous studies.

2.3. Merchantable Whole-Tree Volume

The merchantable whole-tree volume equation (V_t) includes stem volume (v_s or V_s) and merchantable branch volume (V_b). We chose to develop a merchantable whole-tree volume equation instead of a merchantable branch volume equation because merchantable branch volume will be estimated by subtraction once stem volume is estimated up to the merchantable height (h_m). All 64 trees were used for development of the taper equation, but only the 34 trees from Altsasu (summary statistics in

Table 3. Descriptive statistics of the sample of trees used for merchantable whole-tree volume equation (V_t) development.

Variable	Mean	Minimum	Maximum	Std. Dev.
Vt	1.61	0.60	3.57	0.70
D	40.2	27.0	58.5	7.71
H	24.8	20.2	31.4	3.01
dwt	8.5	7.0	10.5	0.76

V_t, merchantable whole-tree volume (m³), including stem volume and merchantable branch volume; D, diameter at breast height (cm); H, total tree height (m); d_wt, whole-tree top diameter (cm), branches are also included.

) were used to develop the merchantable whole-tree volume equation (V_t) because merchantable branch volumes were calculated.

Linear, power and exponential models were fitted during the development of the V_t equation. The linear models were fitted by linear regression using the least squares method, and the non-linear models were fitted by ONLS using the Gauss–Newton iterative method [19]. The initial values used in the iteration process were obtained by linear regression on the logarithmic transformation of the models. This approach is useful for solving many fitting problems in nonlinear models [20]. Diameter at breast height (D), total tree height (H), the combined variable D²H, the merchantable height (h_m), and the h_m/H ratio were tested as independent variables. In the present study, the h_m/H ratio ranged from 0.36 to 0.91, with a mean value of 0.72.

Merchantable whole-tree volume (V_t) should be always greater than or equal to stem volume (V_s). However, this restriction was not included in the model formulation. A simulation was carried out to verify that V_t ≥ V_s. A database of D in the range 10–100 cm was simulated at intervals of 5 cm, and all possible combinations were crossed with a database of H in the range 5–60 m, at intervals of 5 m. V_s was selected in the comparison because it is the highest value estimated by the taper equation.

2.4. Model Fitting

Statistical and graphical analyses were conducted to compare the performance of the equations. Plots of residuals against predicted values and plots of observed values against predicted values were examined to evaluate heteroscedasticity, anomalous trends and bias. Moreover, in development of the taper equation, the residuals were evaluated across relative height and D classes [21], both in absolute values of residuals and in relative values ($\left(Y_i-{\widehat{Y}}_i\right)/Y_i$, see notation below). The numerical analysis consisted of comparing three statistical criteria obtained from the residuals [22]: the coefficient of determination (R²), which measures the amount of the observed variability explained by the model; the mean bias (e), which is the average difference between the values estimated by a model and the actual observed values; and the root mean square error (RMSE), which provides a measure of the accuracy of the estimates in the same units as the dependent variable. Despite the recognized limitations attributed to the utilization of R² in nonlinear regression analysis, it is pertinent to underscore that the broader applicability of a comprehensive measure of model adequacy may supersede such constraints [23]. The mean bias (e) and RMSE are also expressed as percentages (e (%) and RMSE (%)). The expressions of these statistics are summarized as follows:

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^{i=n}{\left(Y_i-{\widehat{Y}}_i\right)}^2}{{\sum }_{i=1}^{i=n}{\left(Y_i-\overline{Y}\right)}^2}}$$

(1)

$$\mathrm{e}={\displaystyle \frac{{\sum }_{i=1}^{i=n}{Y}_i-{\widehat{Y}}_i}{n}}\mathrm{e}(\%)={\displaystyle \frac{\mathrm{e}}{\overline{Y}}}100$$

(2)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^{i=n}{\left(Y_i-{\widehat{Y}}_i\right)}^2}{n-p}}}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}(\%)={\displaystyle \frac{\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}{\overline{Y}}}100$$

(3)

where $Y_i,{\widehat{Y}}_i$, and $\overline{Y}$ are the observed, estimated, and the mean values of the dependent variable, respectively; n is the total number of observations used to fit the model, and p is the number of model parameters.

In regression analysis, the error terms should be independent, identically distributed, normal, random variables. However, in developing taper equations, multiple observations for each tree are used (i.e., longitudinal data). Thus, it is reasonable to expect that the observations within each tree will be spatially correlated and the assumption of independent error terms will therefore be violated [4]. Autocorrelation can be taken into account by modelling the error term. To correct the autocorrelation between residuals from the same tree, we used a continuous autoregressive error structure (CAR(x)), which accounts for the distance between measurements. This error structure expands the error term to [24]:

$$e_{ij}={\sum }_{k=1}^{k=x}{I}_k{\rho }_k^{h_{ij}-h_{ij-k}}{e}_{ij-k}+{\epsilon }_{ij}$$

(4)

where e_ij is the jth ordinary residual of the ith individual, e_ij−k is the j-kth ordinary residual of the ith individual, I_k = 1 when j > k and 0 when j ≤ k, ρ_k is the k-order continuous autoregressive parameter to be estimated, and h_ij-h_ij−k is the distance separating the jth from the j-kth observation within each tree i, with h_ij > h_ij−k. In this case, ε_ij is an independent normally distributed error term with mean value of zero.

To assess the existence of autocorrelation and determine the appropriate order of the continuous autoregressive error structure (CAR(x)), plots of residuals versus lag-residuals (residuals from previous observations) within each tree were visually inspected. Appropriate fits for the selected model with correlated errors were obtained by including the CAR(x) error structure in the MODEL procedure of SAS/ETS^® 9.2 software [25], which allows for dynamic updating of the residuals. In developing the merchantable whole-tree volume equation (V_t), the linear models were fitted using the REG procedure of SAS/STAT^® 9.2 [26] and the non-linear models were fitted using the NLIN procedure of the same software.

3. Results

3.1. Taper Equation

Although all models tested satisfactorily estimated diameters at different heights, the model of Fang et al. [16] performed slightly better than the others, and/or all of its parameters were significant at the α = 5% significance level. This model assumes that the stem has three sections, each with a different form factor:

$$\begin{array}{c}d=c_1\sqrt{H^{\left(z-b_1\right)/b_1}{\left(1-q\right)}^{\left(z-\beta \right)/\beta }{\alpha }_1^{I_1+I_2}{\alpha }_2^{I_2}}\\ \mathrm{where}q=h/H,\mathrm{and}\left\{\begin{array}{c}{I}_1=1\hspace{0.33em}if\hspace{0.33em}{p}_1\le q\le p_2;\hspace{0.33em}0\mathrm{otherwise}\\ I_2=1\hspace{0.33em}if\hspace{0.33em}{p}_2<q\le 1;\hspace{0.33em}0\hspace{0.33em}\mathrm{otherwise}\end{array}\right.\\ \begin{array}{l}{p}_1\mathrm{and}{p}_2\mathrm{are}\mathrm{relative}\mathrm{from}\mathrm{ground}\mathrm{level}\mathrm{where}\mathrm{the}\mathrm{two}\mathrm{inflection}\mathrm{points}\mathrm{are}\\ \mathrm{assumed}\mathrm{by}\mathrm{the}\mathrm{model}\mathrm{to}\mathrm{occur},\mathrm{dividing}\mathrm{the}\mathrm{in}\mathrm{three}\mathrm{sections}\end{array}\\ \beta =b_1^{1-\left(I_1+I_2\right)}{b}_2^{I_1}{b}_3^{I_2}{\alpha }_1={\left(1-p_1\right)}^{{\displaystyle \frac{\left(b_2-b_1\right)z}{b_1{b}_2}}}{\alpha }_2={\left(1-p_2\right)}^{{\displaystyle \frac{\left(b_3-b_2\right)z}{b_2{b}_3}}}\\ r_0={\left(1-h_{st}/H\right)}^{z/b_1}{r}_1={\left(1-p_1\right)}^{z/b_1}{r}_2={\left(1-p_2\right)}^{z/b_2}\\ c_1=\sqrt{{\displaystyle \frac{a_0{D}^{a_1}{H}^{a_2-z/b_1}}{b_1\left(r_0-r_1\right)+b_2\left(r_1-{\alpha }_1{r}_2\right)+b_3{\alpha }_1{r}_2}}}\end{array}$$

(5)

where d is the diameter (cm) at height h; h is the height above ground (m) to diameter d; D is the diameter at breast height (1.3 m above ground, cm); H is the total tree height (m); h_st is the stump height (m); z is π/40,000, a metric constant for converting from diameter squared in cm² to cross-section area in m²; b_i is the form factor of tree section i; and a₀–a₂, b₁–b₃, p₁, and p₂ are parameters to be estimated.

Integration of Equation (5) to any height limit (h) produces the merchantable volume Equation (6), m³. Assuming the tree has an excurrent form, integration of Equation (1) over total tree height (H) produces the stem volume Equation (7), m³.

$$v_s=c_1^2{H}^{z/b_1}\left(b_1{r}_0+\left(I_1+I_2\right)\left(b_2-b_1\right)r_1+I_2\left(b_3-b_2\right){\alpha }_1{r}_2-\beta {\left(1-q\right)}^{z/\beta }{\alpha }_1^{I_1+I_2}{\alpha }_2^{I_2}\right)$$

(6)

$$V_s=a_0{D}^{a_1}{H}^{a_2}$$

(7)

The model of Fang et al. [16] was initially fitted using ONLS, without expanding the error terms to account for autocorrelation. A trend in residuals as a function of lag-residuals within the same tree was apparent, as expected because of the longitudinal nature of the data used for model fitting (Figure 3, first row). To correct autocorrelation, the error term was modelled with a second order CAR structure, CAR(2), since the first order structure, CAR(1), did not totally correct autocorrelation with more than one lag (Figure 3, third and second rows, respectively). Although accounting for autocorrelation does not improve the predictive ability of the taper equation, it prevents underestimation of the covariance matrix of the parameters, thereby enabling use of the usual statistical tests [27], i.e., it allows valid interpretation. All parameter estimates, including autocorrelation parameters, were significant at the 5% level (

Table 4. Parameter estimates and approximate significance tests for the model of Fang et al. [16] (Equation (5)) for Q. rubra in Navarra.

Parameter	Estimate	Approx. Std. Error	t Value	Approx. p-Value
p1	0.06225	0.00609	10.22	<0.0001
p2	0.7590	0.0166	45.68	<0.0001
b1	0.00001172	8.475 × 10−7	13.83	<0.0001
b2	0.00002820	4.258 × 10−7	66.24	<0.0001
b3	0.00005116	8.637 × 10−6	5.92	<0.0001
a0	0.00007717	0.000024	3.27	0.0011
a1	1.815	0.0508	35.77	<0.0001
a2	0.9259	0.0934	9.91	<0.0001
ρ1	0.7435	0.0251	29.60	<0.0001
ρ2	0.7357	0.0205	35.97	<0.0001

).

The sole purpose of correction for autocorrelation was to obtain unbiased and efficient estimates of the parameters [28], and it has no use in practical application. Therefore, to obtain a better idea of the true error expected, for practical use of the taper equation, we calculated the residuals obtained without autocorrelation parameters. The values of the statistical criteria were R² = 0.901, e = 0.369 cm, e (%) = 1.42, RMSE = 3.54 cm and RMSE(%) = 13.7. The plot of residuals against estimated values showed a random pattern of residuals around zero, with no detectable significant trends (Figure 4).

The behaviour of residuals across the diameter at breast height classes was homogeneous both in absolute values and relative values of residuals (Figure 5, first row). The behaviour of residuals in absolute values across the relative height class was also homogeneous (Figure 5, second row, left), but the relative residuals increased with relative height (Figure 5, second row, right).

3.2. Merchantable Whole-Tree Volume

The merchantable whole-tree volume equation (V_t) includes stem volume (v_s) and merchantable branch volume (V_b, with d_min = 7 cm). The values of the statistical criteria for the model selected (Equation (8)) were R² = 0.861, e = −0.00625 m³, e(%) = −0.389, RMSE = 0.270 m³, and RMSE(%) = 16.8.

$$V_t=t_0\mathrm{e}\mathrm{x}\mathrm{p}\left(t_{1}D\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(t_{2}H\right)$$

(8)

where V_t is the merchantable whole-tree volume (m³); D is the diameter at breast height (cm); H the total tree height (m); and t₀, t₁, and t₂ are the fitted parameters, all of which were significant at the 5% level (

Table 5. Parameter estimates and approximate significance tests for Equation (8), which estimates merchantable whole-tree volume (V_t), including stem and merchantable branches (d_min = 7cm) for Q. rubra in Navarra.

Parameter	Estimate	Approx. Std. Error	t Value	Approx. p-Value
t0	0.1520	0.0355	4.29	0.00016
t1	0.04258	0.00352	12.11	<0.0001
t2	0.02363	0.00892	2.65	0.0126

). The plot of residuals against estimated values for Equation (8) showed a random pattern of residuals around zero, with no detectable significant trends (Figure 6).

3.3. System of Equations

Both the taper equation and merchantable whole-tree volume (V_t) equation have been integrated into the Cubica Navarra 3.0 software (https://administracionelectronica.navarra.es/CubicacionMadera/, accessed on 24 September 2024), to be used as a system of equations. The input variables are diameter at breast height (D), total tree height (H), and merchantable height (h_m). If h_m is not specified, the system considers a well-defined excurrent form, with the stem top height defined by the minimum top diameter (d_min) selected by the forest manager. For merchantable whole-tree volume, d_min was fixed at 7 cm. The system also classifies products in the stem, with the remaining merchantable volume assumed to be in the branches, estimated by subtracting the stem volume from the merchantable whole-tree volume. Simulations performed with all possible combinations of D = 10–100 cm (at 5 cm intervals) and H = 5–60 m (at 5 m intervals) produced logical results (V_t > V_s) across the whole range of values.

4. Discussion

We developed a merchantable whole-tree volume model system, including stem and merchantable branches, based on the current measurements available from the Navarra Regional Forest Service. In accordance with Ver Planck and MacFarlane [8], this study did not exclude forked trees or trees of poor form, typically seen in hardwood species, and which would exclude trees with large branches. MacFarlane and Weiskittel [5] describe some defects in the ideal excurrent stem form which lead to trees or even stands being excluded in studies of stem taper equation development. The objective of the present study was to improve economic assessment of wood products, focusing on stem volume, as this prioritizes economic profitability while also considering the volume of merchantable branches.

The data used to develop the present system of equations is not widely representative regarding either the spatial distribution or number of trees; moreover, the ranges of diameters (25–58.5 cm) and heights (17.5–31.4 m) did not include small or overmature trees. However, in developing the taper equation, we used 930 observations of stem diameters along the bole, which is more than the minimum of 825 observations reported by Kitikidou and Chatzilazarou [29] to be needed to parameterize taper equations correctly. We also used 64 trees, which is greater than the minimum 40–85 sample trees needed, and measurements of diameters along the stem were taken at least every 2 m [21]. For development of merchantable whole-tree volume equations, it is advisable to use several stands because tree form varies between stands as a result of several factors such as site productivity, density, genetics, stand management, and historical biotic or abiotic damage [5,21,30,31,32,33]. Taking biomass into account, Montagu et al. [34] associated the proportion of mass allocated to branches with location and D. Stem form and partitioning of biomass among components have been shown to be affected by density, thinning, pruning and fertilization [35]. In this study, only two locations were included; therefore, the developed model system is considered preliminary and independent data from other locations will be useful for refitting or validating it. In the taper equation development, we did not consider the discontinuity of diameter produced by the crown basis [36] or the forks [6]. However, a new variable—already measured by the Navarra Regional Forest Serviceis—is considered, the merchantable height (h_m), simplifying consideration of the rest of the merchantable volume (including the wood above h_m) as branches.

In the prediction of merchantable whole-tree volume, inclusion of the available merchantable height (h_m) or the h_m/H ratio as independent variables did not improve the predictive capacity of the model. MacFarlane [7] found that variables such as crown ratio (CR) and the diameter at the base of the largest branch were good predictors of large branch volume. Crown ratio can be calculated by measuring height to the crown base, in addition to total tree height. Digital hypsometers can be used to measure several heights. However, variables such as diameter at the base of the largest branch or crown diameter are not easily measured. MacFarlane [7] selected D and CR as predictor variables in developing species-specific large branch volume equations. Mäkelä and Valentine [37] also indicated that CR is useful for tree allometry prediction. Crown length can be also used [38]. McTague and Weiskittel [21] mentioned that the exploration of crown variables is encouraged in the development of taper equations. Moreover, the widely used crown classes (dominant, codominant, intermediate, and suppressed trees) could be tested [39]. The use of emerging technologies such as terrestrial laser scanning (TLS) could improve the development and/or utilization of taper and volume equations [21]. Moreover, non-destructive technologies such as ultrasonic and thermal images could be used to detect internal defects [40].

Large branches can be defined as those with a significant impact on stem form [41] or through a minimum top diameter. MacFarlane [7] defined large branches in hardwoods in Michigan state (USA) as those of diameter ≥ 9 cm and observed that the proportion of sampled trees with large branches changed as a function of the D class, ranging from 0% in the 10-cm class up to almost 100% in ≥60 cm classes. In our study, minimum top diameter for merchantable branches is 7 cm, and only one tree (D = 38 cm) did not have merchantable branch volume (V_b). The V_b/V_t ratio ranged from 0 to 43% (mean 20%), and there was a non-significant positive correlation between V_b/V_t and D (Pearsons r = 0.25, p-value = 0.159). A weak positive correlation between V_b/V_t and D was also observed by MacFarlane [7] and Ver Planck and MacFarlane [8]. It should also be taken into account that the data used in this study did not include small trees, and Gómez-García et al. [42] reported that the contribution of each tree biomass component to the total biomass tends to stabilize as the tree diameter increases in Quercus robur L. growing in NW Spain.

Minimum top diameter (d_min) for merchantable volume is not a global standard and includes values of e.g., 5 cm [43], 7 cm [42,44], 9 cm [7,45], and 10 cm [5,46]. Moreover, regarding the stem form and the merchantability of branches, values other than the minimum top diameter are considered, and our data included values of top diameter per whole-tree (stem and branches) in the range 7.0–10.5 cm. On the other hand, new biomass harvesting methods and industrial uses have increased the demand for previously non-merchantable products [47], leading to a change in the minimum top diameter.

More informative models of whole-tree volume could be developed by measuring and testing ancillary variables. For instance, it may be possible to develop equations modelling the vertical allocation of branch volume [8], equations with a variable minimum top diameter [48], and considering different merchantable form types [5]. Such approaches should also include trees with decurrent form. Further research is also necessary to consider variables related to stem form, branch architecture and volume, with the purpose of developing more biologically based models.

5. Conclusions

This study highlights the significance of accounting for both stem and branch volume when estimating merchantable whole-tree volume for Quercus rubra, a decurrent species. The proposed system of equations successfully explained 90% of the observed variability in diameter and 86% in merchantable whole-tree volume. Although autocorrelation correction was applied to the taper equation, it did not enhance predictive accuracy, but it did ensure unbiased parameter estimates, thus improving the reliability of the model.

The Fang et al. taper model proved the most effective for estimating diameters at different stem heights, with all parameters showing statistical significance. Additionally, the merchantable whole-tree volume equation was able to accurately predict both stem and branch volume, with a RMSE of 0.270 m³. This indicates a strong relationship between tree size variables and merchantable whole-tree volume.

These equations have been integrated into the Cubica Navarra 3.0 software for practical application in routine forest inventories. Nevertheless, further validation with a larger dataset covering a broader geographic range and including trees of different age classes and forms is necessary to refine the models and enhance their generalizability.