Compatible Taper and Volume Systems Based on Volume Ratio Models for Four Pine Species in Oaxaca Mexico

Article Compatible Taper and Volume Systems Based on Volume Ratio Models for Four Pine Species in Oaxaca Mexico Albert Castillo-López 1,*, Gerónimo Quiñonez-Barraza 2, Ulises Diéguez-Aranda 3and JoséJavier Corral-Rivas 4   Citation: Castillo-López, A.; Quiñonez-Barraza, G.; Diéguez-Aranda, U.; Corral-Rivas, J.J. Compatible Taper and Volume Systems Based on Volume Ratio Models for Four Pine Species in Oaxaca Mexico. Forests 2021,12, 145. https://doi.org/10.3390/f12020145 Received: 9 December 2020 Accepted: 23 January 2021 Published: 27 January 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Doctorado Institucional en Ciencias Agropecuarias y Forestales, Universidad Juárez del Estado de Durango, Río Papaloapan, Valle del Sur, 34120 Durango, DGO, Mexico 2Campo Experimental Valle del Guadiana, Instituto Nacional de Investigaciones Forestales, Agrícolas y Pecuarias, Carretera Durango-Mezquital km 4.5, 34170 Durango, DGO, Mexico; [email protected] 3 Departamento de Ingeniería Agroforestal, Escuela Politécnica Superior de Lugo, Universidad de Santiago de Compostela, Campus Universitario, 27002 Lugo, Spain; [email protected] 4Facultad de Ciencias Forestales, Universidad Juárez del Estado de Durango, Río Papaloapan, Valle del Sur, 34120 Durango, DGO, Mexico; [email protected] *Correspondence: [email protected]; Tel.: +52-951-218-0065 Abstract: Estimating tree volume components is an essential element in sustainable forest management. Compatible taper and merchantable outside-bark volume systems based on volume ratio models were globally fitted to four pine species (reduced models) with the aim to select the best reduced model and to fit it with dummy variables and additive effects using Pinus pseudostrobus Lindl as a base species. The study was carried out in the northern mountains of Oaxaca, Mexico. To fit the taper, merchantable volume, stem volume, branch volume, and total tree volume (stem volume and branch volume) equations, a taper dataset of 222, 230, 245, and 333 trees of Pinus douglasiana Martínez (Pd), Pinus oaxacana Mirov (Po), Pinus patula Schltd (Pp), and Pinus pseudostrobus Lindl (Pps), respectively, was used. In general, the compatible systems explained more than 97% in the observed variability for the four studied components: outside-bark diameter ( d ), merchantable outside-bark volume ( Vm ), stem outside-bark volume ( Vs ), and total tree volume ( Vt ). Alternatively, more than 52% of the observed variability for branch volume ( Vb ) was also explained. The developed compatible systems based on volume ratio models are a simple and consistent alternative for estimating the outside-bark diameter and variable outside-bark volume, as well as the components of commercial species for uneven-age and mixed-species forests in Oaxaca, Mexico. Keywords: reduced and full compatible system; merchantable outside-bark volume; stem outsidebark volume; total tree volume; branch volume 1. Introduction Sustainable forest management incorporates the use of statistical tools for estimating the total and merchantable volume of trees for the correct application of silvicultural treatments in a specific stand or group of stands, estimating the distribution of commercial products and the economic yield [ 1 ]. Forest species, site quality, stand density, and silvicultural treatments have a direct relationship with the tree profile [ 2 – 4 ] and these elements are used in the formulation of taper and volume systems [1,4,5]. In the last decades, numerous taper models have been proposed, starting with simple hyperbolic expressions, polynomial functions, segmented equations, or spline functions, and directly through volume ratio functions [ 6 , 7 ]. However, several studies have considered that the form of the stem does not correspond to a single equation of dendrometric type, but that the form differs to some extent with the tree species and the point in height along the stem [ 8 ]. Furthermore, the development during recent years of new flexible Forests 2021,12, 145. https://doi.org/10.3390/f12020145 https://www.mdpi.com/journal/forests Forests 2021,12, 145 2 of 15 approaches to derive compatible taper and volume systems to which data corresponding to different species and trees with variable stem forms can be accurately fitted (e.g., [ 1 , 9 – 11 ]). Therefore, this is a good justification for reviewing the appropriateness of compatible taper and volume equations systems derived from volume ratio functions which are based on relative tree height for different tree species. Ratio volume equations estimate the volume of a tree up to a certain diameter or height as a percentage of the total tree volume (stem volume and branch volume) [ 12 – 15 ]. These models consider the fitting of an equation that characterizes the tree form and are mathematical relationships between the diameters or sections of the stem at any point of the tree and the height at which the diameter is estimated [16]. However, up to date most of the research related to the estimation of tree volume has focused on developing compatible tree taper and stem volume models that describe the stem profile and estimate the total or partial stem volume, without considering the volume of branches [ 17 , 18 ]. The total tree volume models that include the branch component are required to provide base information for the sustainable use of the entire tree volume [ 17 , 19 ] because in Mexican forestry the total tree volume is the most important variable for forest management and planning. The suitable estimation of the branch volume is very important because these sections are increasingly used not only for bioenergy purposes but also for quantification of CO 2 sequestration in forest ecosystems [ 20 , 21 ]. The estimation of branch volume is also relevant for assessing the potential bioenergy purposes on the soil nutrient and carbon cycle [22]. The objective of this study was to fit three compatible taper and merchantable outsidebark volume equations as simultaneous systems for estimating the taper (d, cm), merchantable outside-bark volume (Vm, m 3 ), stem outside-bark volume (Vs, m 3 ), branch volume (V b , m 3 ), and total tree volume (V t , m 3 ) components for main pine species in Oaxaca, Mexico. The model fitting was tested for all combined species with a reduced equations system with the same set of global parameters for the four studied tree species and with a full equation system obtained by expanding in the best reduced system each global parameter by including an associated parameter and a dummy variable to differentiate the species and to assess whether separate models are necessary for the selected pine species. 2. Materials and Methods 2.1. Study Area The study was carried out in mixed-species forests in the Regional Forest Management Unit (UMAFOR 2001), in the northeast of Oaxaca state in Mexico. The study area covers an area of 195,397 ha. The UMAFOR 2001 is located at geographic coordinates 17-09 0 56.93 00 — 17-33 0 01.82 00 N and 96-16-42.75 00 —96-22 0 29 00 O (Figure 1). The elevation of the sampling areas ranges between 2031–3361 m. The annual mean temperature varies between 3–28 ◦C, with a mean annual rainfall of 3797 mm [ 23 ]. The predominant vegetation type corresponds to pine and oak species, mainly dominated by Pinus oaxacana Mirov, Pinus douglasiana Martínez, Pinus patula Schltdl, Pinus pseudostrobus Lindl, Pinus leiophylla Schiede, Pinus pringlei Shaw, Pinus rudis Endl, Pinus teocote Schltdl, Quercus crassifolia Bonpl, Quercus laurina Bonpl, and Quercus rugosa Née. Forests 2021,12, 145 3 of 15 Forests 2021, 12, x FOR PEER REVIEW 3 of 15 Figure 1. Location of the study area in Oaxaca, Mexico. 2.2. Data The database considered 3173, 3218, 3660, and 4769 outside-bark diameter measurements at different heights obtained from 222, 230, 245, and 333 trees of Pinus douglasiana Martínez (Pd), Pinus oaxacana Mirov (Po), Pinus patula Schltd (Pp), and Pinus pseudostrobus Lindl (Pps), respectively. All trees were randomly selected in the study area. The diameter, height, site index, and crown class condition were considered. Diameter at breast height (1.3 m above ground level) (dbh; cm) was measured to the nearest 0.1 cm in each tree. The trees were later felled leaving stumps of average height 0.15 m, and total tree height (H; m) was measured to the nearest 0.01 m. The trees were cut into bolts as follows. The first three bolts were of the constant length of 0.3 m and the third log was of variable length because the upper diameter coincided with dbh. These measurements were considered to describe the tree profile in the lower section of each tree in which taper changes so rapidly low on the bole. Subsequent bolts were 2.54 m long. All tree branches, defined as those with a diameter at base greater than or equal to 5 cm, were also sectioned at variable lengths, those lengths ranged from 1.0 to 10 m. The minimum and maximum perpendicular outside-bark diameters were measured in each cross-section (of trunk or branch) to the nearest 0.1 cm and then the arithmetically average was registered. The axes for these measurements depended on the stem form. The volumes of the stem bolts and branches were calculated with the Smalian formula and the top with the cone equation. Table 1 shows the descriptive statistics including number of trees and observations (n), mean, minimum and maximum values, and standard deviation (SD) for the main tree variables of the studied tree species. Figure 2 shows the relative volume versus relative height for the species under study. Figure 1. Location of the study area in Oaxaca, Mexico. 2.2. Data The database considered 3173, 3218, 3660, and 4769 outside-bark diameter measurements at different heights obtained from 222, 230, 245, and 333 trees of Pinus douglasiana Martínez (Pd), Pinus oaxacana Mirov (Po), Pinus patula Schltd (Pp), and Pinus pseudostrobus Lindl (Pps), respectively. All trees were randomly selected in the study area. The diameter, height, site index, and crown class condition were considered. Diameter at breast height (1.3 m above ground level) (dbh; cm) was measured to the nearest 0.1 cm in each tree. The trees were later felled leaving stumps of average height 0.15 m, and total tree height (H; m) was measured to the nearest 0.01 m. The trees were cut into bolts as follows. The first three bolts were of the constant length of 0.3 m and the third log was of variable length because the upper diameter coincided with dbh. These measurements were considered to describe the tree profile in the lower section of each tree in which taper changes so rapidly low on the bole. Subsequent bolts were 2.54 m long. All tree branches, defined as those with a diameter at base greater than or equal to 5 cm, were also sectioned at variable lengths, those lengths ranged from 1.0 to 10 m. The minimum and maximum perpendicular outside-bark diameters were measured in each cross-section (of trunk or branch) to the nearest 0.1 cm and then the arithmetically average was registered. The axes for these measurements depended on the stem form. The volumes of the stem bolts and branches were calculated with the Smalian formula and the top with the cone equation. Table 1shows the descriptive statistics including number of trees and observations (n), mean, minimum and maximum values, and standard deviation (SD) for the main tree variables of the studied tree species. Figure 2shows the relative volume versus relative height for the species under study. Forests 2021,12, 145 4 of 15 Table 1. Descriptive statistics of the studied pine tree species. Species Variable Trees nMinimum Mean Maximum SD Pd dbh (cm) 222 3173 8.95 39.02 74.00 13.46 H (m) 7.40 25.11 38.98 6.24 d(cm) 0.00 26.16 84.60 16.16 h(m) 0.08 11.27 38.98 9.48 Vm(m3)0.0000 0.9943 5.9168 1.0681 Vs(m3)0.0360 1.6618 5.9168 1.1787 Vb(m3)0.0005 0.0230 0.3267 0.0382 Vt(m3)0.0402 1.6848 6.1381 1.2044 Po dbh (cm) 230 3218 8.55 32.31 70.50 11.92 H (m) 7.84 25.69 40.61 6.04 d(cm) 0.00 21.64 77.25 13.72 h(m) 0.03 11.08 40.61 9.39 Vm(m3)0.0000 0.6536 5.7910 0.8069 Vs(m3)0.0363 1.1058 5.7910 0.9693 Vb(m3)0.0005 0.0135 0.1441 0.0199 Vt(m3)0.0405 1.1192 5.9058 0.9837 Pp dbh (cm) 245 3660 9.60 41.54 80.50 15.23 H (m) 7.10 29.94 49.28 7.62 d(cm) 0.00 28.14 88.50 16.91 h(m) 0.05 13.21 49.28 11.04 Vm(m3)0.0000 1.3366 8.9478 1.5338 Vs(m3)0.0242 2.2816 8.9478 1.8388 Vb(m3)0.0003 0.0252 0.1759 0.0301 Vt(m3)0.0284 2.3068 9.0721 1.8628 Pps dbh (cm) 333 4769 8.30 39.60 81.15 14.63 H (m) 9.36 28.18 45.55 6.86 d(cm) 0.00 27.09 91.50 16.68 h(m) 0.07 12.28 45.55 10.39 Vm(m3)0.0000 1.1356 9.2150 1.3631 Vs(m3)0.0425 1.9719 9.2150 1.6818 Vb(m3)0.0004 0.0267 0.3876 0.0390 Vt(m3)0.0467 1.9986 9.3117 1.7080 n= number of observations; SD = standard deviation; dbh = diameter at breast height (cm); H= total tree height (m); d= outside-bark diameter at height h(cm); h= upper-height measure from the ground to the top tree height (m); Vm = merchantable outside-bark volume (m 3 ); Vs = stem outside-bark volume (m 3 ); Vb = branch volume (m 3 ); Vt= total tree stem volume (m3). 2.3. Total Tree Volume Equation In the forest literature, there are different volume equations to predict tree volume, including the equation of Schumacher and Hall [ 24 ], which has been widely used for different commercial species [ 1 , 3 , 25 , 26 ], and therefore it was selected to estimate the stem outside-bark volume ( Vs ) in this study (Equation (1)). To predict the branch volume ( Vb ), the power model was used with a scaled intercept parameter on the exponential function (Equation (2)). Equation (2) is a modification of Equation (1), but the parameter of height variable was not significantly different to zero at 5% of significance level. Vs=α0dbhα1Hα2(1) Vb=eγ0dbhγ1(2) where V s = stem outside-bark volume (m 3 ); dbh = diameter at breast height (cm); Vb= branch volume (m 3 ); H= total tree height (m); αi and γi = parameters to be estimated through regression analysis. Forests 2021,12, 145 5 of 15 Forests 2021, 12, x FOR PEER REVIEW 5 of 15 (a) (b) (c) (d) Figure 2. Relative volume (𝑉/𝑉) and relative height (h/H) for Pinus douglasiana (Pd; a); Pinus oaxacana (Po; b); Pinus patula (Pp; c); and Pinus pseudostrobus (Pps; d). 2.3. Total Tree Volume Equation In the forest literature, there are different volume equations to predict tree volume, including the equation of Schumacher and Hall [24], which has been widely used for different commercial species [1,3,25,26], and therefore it was selected to estimate the stem outside-bark volume (𝑉) in this study (Equation (1)). To predict the branch volume (𝑉), the power model was used with a scaled intercept parameter on the exponential function (Equation (2)). Equation (2) is a modification of Equation (1), but the parameter of height variable was not significantly different to zero at 5% of significance level. 𝑉=𝑒 𝑑𝑏ℎ (2) where Vs = stem outside-bark volume (m3); dbh = diameter at breast height (cm); Vb = branch volume (m3); H = total tree height (m); αi and 𝛾i = parameters to be estimated through regression analysis. The parameters of Vs equation correspond to αi (𝑖=0,1,2) were fully compatible with taper (d) and merchantable volume (Vm) equations. The total tree volume (Vt) was modeled as the sum of Vs and Vb components (Equation 3). 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 V m /V s h/H Pd 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 V m /V s h/H Po 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 V m /V s h/H Pp 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 V m /V s h/H Pps 𝑉=α0𝑑𝑏ℎ𝐻 (1) Vt=α0𝑑𝑏ℎ𝐻+𝑒𝑑𝑏ℎ (3) Figure 2. Relative volume ( Vm / Vs ) and relative height (h/H) for Pinus douglasiana (Pd; a ); Pinus oaxacana (Po; b ); Pinus patula (Pp; c); and Pinus pseudostrobus (Pps; d). The parameters of V s equation correspond to αi(i=0, 1, 2) were fully compatible with taper (d) and merchantable volume (Vm) equations. The total tree volume (V t ) was modeled as the sum of Vsand Vbcomponents (Equation (3)). Vt=α0dbhα1Hα2+eγ0dbhγ1(3) 2.4. Compatible Taper and Volume Systems Derived from Ratio Functions Based on Relative Height The ratio volume models that depend on the relative height were studied by Broad and Wake [ 7 ], Cao and Burkhart [ 27 ], Cao, Burkhart and Max [ 14 ], Parresol, et al. [ 28 ], Bullock and Burkhart [ 29 ], and most recently by Zhao and Kane [ 9 ], who used biomass and volume functions for the ratio of commercial height. Van Deusen et al. [ 30 ] and Reed and Green [ 15 ] derived taper models compatible with volume ratio functions based on total height. The function is R(p)=Vm Vs , Vm (m 3 ) corresponds to the cumulative volume of the stem from the base to the height h(m) above the ground, where Vs (m 3 ) is the total stem volume and p=h H is the ratio between the merchantable (h) and total height (H). Zhao and Kane [ 9 ] and Lynch, Zhao, Harges, and McTague [ 10 ] proposed four conditions that ratio volume functions must meet are (I) R= 0, if p= 0, (II) R= 1, if h=H , (III) ∂R/∂h≥ 0, for 0 ≤p≤ 1 y (IV) ∂2R/∂2p≤ 0, for 0 ≤p≤ 1, also these properties were satisfied by Quiñonez-Barraza, Zhao, and De los Santos-Posadas [ 1 ] in compatible taper and volume equations for pine species in Durango, Mexico. The compatible taper and volume equations systems using in this study are shown in Table 2. In a preliminary analysis, these three systems performed better than others. These systems were arranged for d , and Vm components and were called CS1, CS2, and CS3 [ 9 , 10 ]. These systems are compatible with Vs,Vb, and Vtcomponents in Equations (1)–(3). Forests 2021,12, 145 6 of 15 Table 2. Compatible taper and merchantable volume systems based on volume ratio models fitted to studied species. System Equation CS1 Vm=α0dbhα1Hα2h Hexphβ01−h Hiεij; 0 <β0≤1, 0 ≤h H≤1 d=hα0dbhα1Hα2 Hk exphβ01−h Hi1−β0h Hi0.5 +εij CS2 Vm=α0dbhα1Hα21−1−h Hβ0exp−β1h Hεij;β0≥1, β1>0 , 0 ≤h H≤1 d=α0dbhα1Hα2 Hk exp−β1h H1−h Hβ0−1hβ0+β11−h Hi0.5 +εij CS3 Vm=α0dbhα1Hα21−1−h Hβ0(1−β1exp(−exp(β2dbhα1Hα2))) εij;β0≥0 , 0 ≤h H≤1 d=α kH α0dbhα1Hα21−h Hβ0−1(1−β1exp(−exp(β2dbhα1Hα2)))1−1−h Hα(−β1exp(−exp(β2dbhα1Hα2)))0.5 +εij Vm = merchantable outside-bark volume from the base to the height h(m); dbh = diameter at breast height (cm); H= total tree height (m); h = upper height (m); d= upper diameter at height h; k = π /40,000 for metric units; αi , βi = parameters to be estimated; εij = error of jth measurement in the ith tree. The used dummy variables for the full model were considered as additive effects to Pinus pseudostrobus (Pps) because this was the most abundant and representative tree species in the dataset. The dummy variables were expressed following the method studied by Quiñonez-Barraza et al. [ 31 ]; Ij=1 if sp =j 0 otherwise , where Ij represents the dummy variable for species (sp); j= 2 for Pd, j= 3 for Po, and j= 4 for Pp. The global parameters were rewritten based on dummy variables, so that αi and βi could be represented as αi=αi1+αi2I2+αi3I3+αi4I4 and βi=βi1+βi2I2+βi3I3+βi4I4 . The full model with dummy variables only considered the significant parameters different from zero at significance level of 1% (α=0.01) in modeling process. 2.5. Model Fitting, Autocorrelation, Heteroscedasticity, and Equations Comparison The components of the compatible taper and volume systems were simultaneously fitted using the iterative seemingly unrelated regression technique (ITSUR) [ 32 ] with MODEL procedure of the SAS/ETS ® [ 33 ]. This technique generates an accurately estimated parameter when the error components are correlated in taper or merchantable volume equations for a specific system. The compatibility between the taper and volume equations allowed the estimated parameter to be the same for each CS and these guaranteed optimizations on fitting process by ITSUR technique [1,31,34,35]. To guarantee compatibility by degrees of freedom between the taper and the volume equations ( Vs , Vb , Vt ), we used the ratio of one over the number of sections of each tree (ni), which generated the weight variable (w = 1/ni) [11,31,36,37]. This procedure was incorporated into the SAS script as resid. ( Vs , Vb , Vt ) = resid.( Vs , Vb , Vt ) √w for Vs , Vb and Vt , respectively. This process allowed to fit the stem volume, total tree volume, and branch volume equations with the same dataset of the taper outside-bark merchantable volume. The main problems associated with taper and volume equations that do not satisfy the fundamental assumptions of regression analysis are autocorrelation and heteroscedasticity [ 38 ]. To correct the autocorrelation of the error term and obtain efficient model parameters [ 39 – 41 ], a second-order continuous autocorrelation structure was used (CAR2) [ 42 – 44 ]. eij=d1ρ1hij−hij−1eij−1+d2ρ2hij−hij−2eij−2+εij (4) where e ij is the jth ordinary residual on the ith individual tree; d1= 1 for j> 1; d2= 1 for j> 2; d1= 0 for j= 1; d2= 0 for j≤ 2; h ij -h ij-1 is the distance separating the jth from the jth-1 observations within each tree; h ij -h ij-2 is the distance separating the jth from the jth-2 observations within each tree; and ρ1 and ρ2 , are the first- and second-order autoregressive parameters. Forests 2021,12, 145 7 of 15 To ensure homoscedasticity in the volume equations, we used a power function of the residual variance ( σi2=dbh2H∅ ), the value of ∅ was estimated by the method proposed by Harvey [ 45 ], using the errors of the fitted equation as dependent variable. The parameters were estimated with the arrangement resid.(Vm,Vs,Vb,Vt)=resid.(Vm,Vs,Vb,Vt) /hdbh2H∅i0.5 for V s ,V t, and V b , respectively [ 1 , 46 ]. The fixed ∅0s parameter were 1.9423, 1.2813, 1.3214, and 1.8507 for Pd,Po,Pp, and Pps, respectively. The evaluation and selection of the most efficient compatible system based on volume ratio equations was performed with the known fit statistics as the adjusted coefficient of determination ( R2adj ), the root mean square error (RMSE), the Akaike’s information criterion (AIC), and the average error ( E ). In addition, graphic analyses of the residuals were examined to identify point for which the systems provide especially poor or good predictions [47]. Fit statistics are defined by the following expressions: R2adj =1−"n−1∑n i=1(yi−yi)2 n−p−1∑n i=1(yi−y)2#(5) RMSE ="∑n i=1(yi−ˆ yi)2 n−1#0.5 (6) AIC =n ln"∑n i=1 (yi−ˆ yi)2 n#+2p(7) E=∑n i=1 (yi−ˆ yi) n (8) where yi , ˆ yi , and y are the observed, predicted, and average values of the dependent variable ( d , Vm , Vs , Vb , and Vt ); nis the number of observations; p is the number of parameters for each equation. 3. Results 3.1. Reduced Compatible Systems Table 3shows the estimated parameters and the fitting statistics of three reduced compatible taper and volume systems (RCSs) derived from height-based ratio equations analyzed in this study for all combined species. All parameters were significantly different to zero at 1% of the significance level. The systems considered using of a second-order continuous-time autoregressive error structure (CAR2) to correct the autocorrelation of the errors and a power function for heteroscedasticity correction in volume equations ( Vm , Vs , Vb , Vt) . Overall, the three compatible systems accounted for 97% of the variance of d , Vm , Vs , and Vt , and 50% of the variance of V b .The RMSE values ranged between 1.75 and 2.29 cm for diameter, from 0.17 to 0.19 m 3 for merchantable outside-bark volume, from 0.17 to 0.24 for V s , from 0.0221 to 0.0222 for V b , and lately from 0.253 to 0.254 for corresponding V t . The RCS3 offered the best fit statistics in R2adj , RMSE , and E in the components d,Vmand Vsand slightly better in the components Vband Vt. Figure 3illustrates the trend of residuals in box and whisker of dand V m components for each reduce compatible system (RCS1, RCS2, RCS3) by relative height classes (h/H; %). The RCS3 presented better accuracy than RCS1 and RCS2 in the different relative height categories (i.e., both dand V m residuals are closer to 0 in comparison to RCS1 and RCS2). Thus, the RCS3 was proposed as the most suitable for describing the stem profile and predicting volume component of the four pine studied species. Forests 2021,12, 145 8 of 15 Table 3. Parameter estimates and fit statistics for the reduced compatible systems analyzed. System Parameter Estimator SE t-Value Pr > |t| Component R2adj RMSE AIC E RCS1 α00.000065 0.000001 64.83 <0.00001 d0.980 2.2940 24617.8 0.259 α11.972095 0.004204 469.06 <0.00001 Vm0.977 0.1919 −48929 −0.052 α20.848143 0.005625 150.77 <0.00001 Vs0.972 0.2472 −2871.4 −0.005 β00.981334 0.001591 616.85 <0.00001 Vb0.509 0.0222 −7831.3 0.001 γ0−15.434798 0.112485 −137.22 <0.00001 Vt0.971 0.2539 −2814.3 −0.005 γ13.087980 0.027454 112.48 <0.00001 ρ10.888375 0.003789 234.45 <0.00001 ρ20.315585 0.007199 43.84 <0.00001 RCS2 α00.000065 0.000001 64.38 <0.00001 d0.980 2.2735 24352.8 0.185 α11.971378 0.004209 468.43 <0.00001 Vm0.978 0.1902 −49190 −0.048 α20.849937 0.005626 151.07 <0.00001 Vs0.972 0.2472 −2871.4 −0.005 β01.659596 0.017481 94.94 <0.00001 Vb0.509 0.0222 −7831.3 0.001 β11.039716 0.021714 47.88 <0.00001 Vt0.971 0.2539 −2814.2 −0.004 γ0−15.432212 0.112549 −137.12 <0.00001 γ13.087266 0.027468 112.39 <0.00001 ρ10.889772 0.003760 236.65 <0.00001 ρ20.326598 0.007468 43.73 <0.00001 RCS3 α00.000061 0.000001 64.54 <0.00001 d0.988 1.7566 16707.8 0.019 α11.975390 0.004096 482.27 <0.00001 Vm0.981 0.1749 −51673.1 −0.021 α20.859596 0.005357 160.47 <0.00001 Vs0.972 0.2474 −2869.6 0.005 β02.016540 0.005936 339.70 <0.00001 Vb0.509 0.0221 −7831.2 0.001 β20.380476 0.003884 97.95 <0.00001 Vt0.971 0.2541 −2812.2 0.005 β30.000002 0.000000 15.19 <0.00001 γ0−15.529946 0.113751 −136.53 <0.00001 γ13.109574 0.027748 112.07 <0.00001 ρ10.919566 0.004373 210.30 <0.00001 ρ20.819601 0.004317 189.84 <0.00001 SE = the standard error of the parameter; t-value = the value of the student’s tdistribution; Pr > |t| = the probability value associated with student’s t-distribution; R2adj = adjusted coefficient of determination; RMSE = root mean square error, AIC = Akaike’s criterion information, E= absolute mean error. *Noted that gamma parameters represent the branch equation (Equation (2)). 3.2. Full Compatible System Table 4shows the estimated parameters and fitting statistics for the full compatible system (FCS3) based on RCS3 for the five components d , Vm , Vs , Vb , and Vt , after expanding each parameter by including an associated parameter and a dummy variable to differentiate among species. Results of dummy variables indicate that there were differences among the compatible taper and volume systems from different pine species (i.e., all the associated parameters were different from zero (p< 0.0001)). For the taper equation, the RMSE was 1.74 cm, while for components Vm , Vs , Vt , and Vb, RMSE values were 0.168, 0.234, 0.24119 and 0.02178 m 3 , respectively. R2adj values were greater than 97.4% for d, Vm , Vs , and Vt equations, and of 52.8% for the Vb equation. The full model indicated that estimated αi0s are different for all species and just for estimated β00s and β20s some parameters are the same. Thus, Pps and Po share the same β0 parameter, while, Pps,Po, and Pd share the same β2parameter. Figure 4shows the trend of residuals in box and whisker of d and V m components by relative height (h/H) of the FCS3 system for the pine species studied. According to Figure 4, FCS3 described well the data for both outside-bark diameter along the stem and cumulative stem volume from the base to the height h. Forests 2021,12, 145 9 of 15 Forests 2021, 12, x FOR PEER REVIEW 9 of 15 The RCS3 presented better accuracy than RCS1 and RCS2 in the different relative height categories (i.e., both d and V m residuals are closer to 0 in comparison to RCS1 and RCS2). Thus, the RCS3 was proposed as the most suitable for describing the stem profile and predicting volume component of the four pine studied species. (a) (b) (c) (d) (e) (f) Figure 3. Distribution of residuals of 𝑑 and 𝑉  of the reduced compatible systems RCS1 (a,b), RCS2 (c,d), and RCS3 (e,f); “o” is the mean, “-” is the median, and “I” is the error bar representing the interquartile range (Q1–Q3). 3.2. Full Compatible System Table 4 shows the estimated parameters and fitting statistics for the full compatible system (FCS3) based on RCS3 for the five components 𝑑, 𝑉, 𝑉, 𝑉, and 𝑉, after expanding each parameter by including an associated parameter and a dummy variable to differentiate among species. Results of dummy variables indicate that there were differences among the compatible taper and volume systems from different pine species (i.e., all the associated parameters were different from zero (p < 0.0001)). For the taper equation, the RMSE was 1.74 cm, while for components 𝑉, 𝑉, 𝑉, and Vb, RMSE values were 0.168, 0.234, 0.24119 and 0.02178 m 3 , respectively. 𝑅 values were greater than 97.4% for d, Figure 3. Distribution of residuals of d and Vm of the reduced compatible systems RCS1 ( a , b ), RCS2 ( c , d ), and RCS3 ( e , f ); “o” is the mean, “-” is the median, and “I” is the error bar representing the interquartile range (Q1–Q3).