Data on volume and taper

The provenance trial was conducted on 1.8 ha study of the INIFAP experimental station El Palmar located in the municipality of Tezonapa, Veracruz, México (18° 32’ 55 N, and 96° 47’ 23 W and 180 m altitude) (Table 1). The climate is hot humid with an annual precipitation of 2885 mm, mean temperature of 24.4 °C, and minimum and maximum temperatures of 16.1 °C and 35.4 °C. Flatlands and hills with slopes of 5 to 29 % make up the physiography. Soils are deep acrisols and vertisols with good natural drainage, crumbly textured sandy clay and pH 4.8 to 6.1 (^{Sánchez and Velázquez, 1998}).

Information was obtained from 211 sixteen-year-old trees, which were the product of the only thinning applied to the progeny trial in December 2010 to establish a seed orchard with 126 individuals per hectare, as suggested by ^{Zobel and Talbert (1988)}. Thinning covered the highest possible diversity in tree sizes and forms to eliminate dominant, co-dominant, intermediate and suppressed individuals. The data were obtained by stem analyses, recording the variables diameter at breast height (cm, Dbh), total height (m, H ), diameter at different stem heights with bark (cm, d ), and heights above ground level of each section (m, h). The first of these measurements was taken at 0.25 m, the second at 1.05 m (Dbh) and the following measurements were every 2 m until reaching the minimum diameter of 3 cm. The register included 1670 pairs of d and h data. For cubage calculations, we used the Overlapping Bolt Method, proposed by ^{Bailey (1995)}, which obtains precise total and partial volumes (^{Cruz et al., 2008}). Figure 1 presents data on taper and accumulated volume.

Figure 1 A) Partial heights in function of partial diameters, and B) commercial volume in function of stem height, for sampled trees. 

Comparison of provenances by form factor

The ^{Spurr (1952)} combined variable model estimates the tree form coefficient, parting from the fact that an irregular shaped body (tree stem form) is commonly compared with a cylinder. It is assumed that the volume of the stem is proportional to the volume of a cylinder (V _a ∝V _c ) and that diameter at breast height (Dbh in cm) and total height (H in m) of the cylinder and the tree are the same.

The volume correlates with Dbh2 and H. In this context, the volume of the cylinder is

However, tree volume will always be smaller than the volume of the theoretical cylinder (V _a <V _c ), and therefore, the tree volume should be proportional to the volume of the cylinder multiplied by a constant (V _a ∝V _c ×C ) smaller than 1 (C<1). Here, C is the tree form factor. This leads us to a first model in which tree volume will be approximately equal to the cylinder volume because of its proportionality constant, such that:

Then

This evidence leads to the expression that represents the model of the constant morphic coefficient, assuming a zero intercept, in the following way:

Eventually, for trees, there is a minimum measurable diameter and any individual with a smaller diameter is assumed to have zero volume, originating the following combined variable model:

where α reflects the effect of the minimum measurable diameter and  will be the form constant as long as the constant k=π /40000 is considered.

To detect differences in intercepts and form factors among provenances in the combined variable model with the direct form factor (Equation 5), a test of simple additionality was developed using a linear regression analysis and incorporating indicator variables (dummies) to denote the absence or presence of some additive effect (^{Montgomery et al., 2005}; ^{Sheng et al., 2011}). Dummy variables were incorporated into the provenances, where α= α ₀ +α ₁ Z ₁ +α ₂ Z ₂ +α ₃ Z ₃ +α ₄ Z ₄ +α ₅ Z ₅ +α ₆ Z ₆ and β= β ₀ + β ₁ Z ₁ + β ₂ Z ₂ + β ₃ Z ₃ + β ₄ Z ₄ + β ₅ Z ₅ + β ₆ Z ₆

In the remainder of the document, if it is not the base provenance of Tuxtepec, then:

Z1=1, if it belongs to Tierra Blanca0,otherwise

Z2=1, if it belongs to Tezonapa0, otherwise

Z3=1, if it belongs to San Andrés Tuxtla0, otherwise

Z4=1, if it belongs to María Lombardo0, otherwise

Z5=1,if it belongs to Costa del Golfo0, otherwise

Z6=1, if it belongs to Comala0, otherwise

where Va is the volume of the entire tree stem (m3); Dbh is the diameter at breast height (cm); H is the total height (m); Z ₁, Z ₂, Z ₃, Z ₄, Z ₅, Z ₆ are the dummy variables; α₀ and β₀ are the regression parameters; α₁, α₂, α₃, α₄, α₅, α₆, β₁, β₂, β₃, β₄, β₅, β₆ are the regression parameters due to additionality and ε the random error. The effects take Tuxtepec as the reference provenance, selected because it has the highest number of observations. The hypothesis considers homogeneity among intercepts and form factors (i.e., H ₀= α₁=α₂=α₃=α₄,=α₅=α₆=β₁=β₂=β₃=β₄=β₅=β₆=0). The procedure for estimating the parameters was by ordinary least squares.

Comparison of provenances by taper

To determine the differences in taper among provenances, the ^{Clutter (1980)} type model of pure taper was taken as the base and expressed as:

Nevertheless, Equation (6) expresses stem form in two components, that is, in the scale fraction of the total stem volume (α Dn ^γ H ^θ ) and in the accumulated volume fraction (H −h)^δ. It is here that α and δ can be considered the most influential parameters in determining total and commercial volume. Therefore, they were restated in function of the dummy variables in such a way that α= α₀+ α₁Z₁+ α₂Z₂+ α₃Z₃+ α₄Z₄+ α₅Z₅+ α₆Z₆ and δ=δ₀+δ₁Z₁+δ₂Z₂+δ₃Z₃+δ₄Z₄+δ₅Z₅+δ₆Z₆. The procedure for estimating the parameters was ordinary least squares.

Where d is the diameter at a partial height h (cm), the rest of the variables have the same notation. The additive effects likewise have the Tuxtepec provenance as base. The working hypothesis poses homogeneity in the taper parameters (i.e. H ₀= α₁=α₂=α₃=α₄,=α₅=α₆=δ₁=δ₂=δ₃=δ₄=δ₅=δ₆=0).

Commercial volume models

Three ^{Fang and Bailey (1999)} type systems were tested to estimate commercial volume and taper simultaneously. These models permit comparing stem form among provenances in a more complete way, that is, in two form parameters. To this end, we replaced the total volume model of Schumacher-Hall, implicit in the systems, with the combined variable model with the direct form factor in β. Also, the taper function and the structure of the volume of the non-commercial tip share another form parameter δ (^{Fang and Bailey, 1999}). Therefore, these two overall parameters were restated in function of the dummy variables to define the additive effect of the different provenances of the trees (Table 2). ^{Corral et al. (2007)} and ^{Quiñonez et al. (2014)} used a similar approach in overall adjustments considering dummy variables for different species of pine. Like the additionality test of the form factors and pure taper, we used the same base provenance and description of the dummy variables.

Table 2 Compatible systems fit to the Spanish cedar provenances established in Tezonapa, Veracruz, Mexico. 

Vc is the commercial volume (m³); d is diameter at partial height h (cm); h is the partial height of the stem (m); H is the total height of the stem (m); k=π/40000; α, β, δ, θ are the regression parameters; Z ₁, Z ₂, Z ₃, Z ₄, Z ₅, Z ₆ are dummy variables for each progeny; β₁, β₂, β₃, β₄, β₅, β₆, δ₁, δ₂, δ₃, δ₄, δ₅, δ₆: are the regression parameters due to additionality.

Fitting strategy

Parameter estimation for the ^{Fang and Bailey (1999)} systems, case I-a, case II-a, case II-b, was under SUR estimation. Initial adjustment tests of the simplest system of equations (case I-a) suggest evidence of self-correlation and heteroscedasticity under SUR and that these increase variance of the system, while the scale of standard errors of the parameters increase very little. ^{Quiñonez et al. (2014)} recommends this fitting procedure when the overall parameters are restated in function of dummy variables that absorb a large part of the system variability, and an inadequate correction of self-correlation and heteroscedasticity could increase the bias of the estimated parameters.

SUR estimation is appropriate for simultaneous equation fitting, to minimizes standard error and to increase the level of significance of the estimators in both the taper equation and commercial volume, increasing hypothesis test sensitivity in the form parameters (^{Rose and Lynch, 2001}). We followed the MODEL procedure of SAS/ETS^* (^{SAS Institute Inc., 2004}) to adjust the models.

To compare the systems and select the best goodness of fit, we used the fitted coefficient of determination (R ² _Adj ), which measures the variability explained by the taper and commercial volume model, the root mean squared error (RMSE), which provides the measure of the average differences between estimated and observed values, the coefficient of variation (CV %), which explains variability relative to the mean response of the dependent variable and the average absolute (S) bias and provides information on the tendency of the model to under-or overestimate the response variable (^{Diéguez et al., 2003}; ^{Galán et al., 2008}; ^{Quiñones et al., 2014}). These statistics are calculated with the following expressions:

where y _i , yi^ and y¯i are the observed, estimated and measured values, respectively, of the response variable; n is the number of observations and p is the number of parameters in the system.

Provenances and form factor

The test comparing groups of provenances by addition to Tuxtepec (Table 3) indicates that all of the 16-year-old progenies have the same intercept to the origin α1=α2=α3=α4=α5=α6, p>0.05), since the parameter estimators are not significantly different from zero at a 0.05 probability of rejection. Regarding the form factors, the Tuxtepec base provenance has slenderer stems, nearly paraboloid (ff=0.44). In this respect, we found that the geographical sources Tierra Blanca β₀= β ₁, p>0.05), Tezonapa β ₀= β ₂, p>0.05) and San Andrés Tuxtla (β ₀= β ₃, p>0.05) were not statistically different, whereas María Lombardo, Costa del Golfo and Comala are different because additionality in the form parameters is statistically significant (β ₄≠ β ₅≠β ₆, p≤0.05).

The negative value of the additionality parameters β₁, β₂, β₃, β₄, β₅ and β₆ (Table 3) is due to the effect of the provenances and indicates that the form factor in the geographic sources of Tuxtepec, Tierra Blanca, Tezonapa and San Andrés Tuxtla is similar, tending toward paraboloid. In contrast, the provenances Costa del Golfo, María Lombardo and Comala, even when the diameters at breast height and total height are the same, are much more conical.

Up to a certain point, form factors can be considered a phenotypical characteristic peculiar to the provenances, the result of genetic potential or “heritability” and the environment in which they grow -soil, wind, moisture, climate, precipitation-as well as of the effect of the pest Hypsipyla grandella Zeller (^{Zobel and Talbert, 1988}).

Provenances and taper

The search for evidence to show the different stem forms among provenances was more sensitive in the additionality test with the ^{Clutter (1980)} pure taper model. The results (Table 4) indicate that the geographical sources Tierra Blanca, Tezonapa, San Andrés Tuxtla, María Lombardo, Costa del Golfo and Comala, by addition to Tuxtepec, have different tapers. That is, there are significant differences in the parameter of the volume scale fraction of the entire stem (α1≠α2≠α3≠α4≠α5≠α6, p≤0.05) and the parameter of the accumulated volume fraction (δ1≠δ2≠δ3≠δ4≠δ5≠δ6, p≤0.05); thus, the effect of the dummy variable Z was significant.

In the ^{Clutter (1980)} type model for taper, it can be observed that all the estimators of the additionality parameters are significantly different from zero. This indicates that, unlike the first approach based on total tree form factor (with a single observation per individual), all the provenances have different tapers relative to the base provenance (Tuxtepec). Given the fact that average Dbh was 13.5 cm and average H was 11 m, when calculating the total stem volume by provenance with the Schumacher-Hall model, derived from ^{Clutter (1980)}, it is estimated that Tuxtepec had the highest form factor (ff=0.57). Moreover, when the taper model is used to estimate height at the 5 cm diameter limit, Tuxtepec exhibits the greatest height limit and commercial volume (Figure 2), and therefore, is the most cylindrical provenance.

Figure 2 Taper variation in Spanish cedar provenances derived from the ^{Clutter (1980)} type model. 

Taper is a phenotypical characteristic resulting from genetic potential, the environment and their interaction, but the most geometrically cylindrical stems from Tuxtepec, Tierra Blanca and Tezonapa reflect the capacity of resistance (tolerance) to or recovery from attack by the borer Hypsipyla grandella Zeller in their juvenile stage. This pest bores the apical meristem, making galleries in the young stems and modifying their morphology (^{Briceño, 1997}). ^{Zepahua and Sánchez (2013)} studied the incidence of the pest in clones of the same progenies and concluded that these geographical sources can tolerate the pest, showing strong apical growth and rapid regeneration of their tissues during attack. In this way, they retain the economic value of their stem. Although this characteristic has great potential value, the genetic and physiological base of the pattern is not entirely understood (^{Newton et al., 1995}). Another interesting factor is natural pruning. It is likely that these provenances can strongly self-prune the lower part of their stems. Since the branches are photosynthetically active, self-pruning the lower branches allows for a higher concentration of chlorophyll in the upper part of the stem where diametric growth increases, resulting in more cylindrical stems (^{Barrio, 2009}).

The ^{Clutter (1980)} type taper model reveals that all the provenances are different, possibly because in the ^{Spurr (1952)} combined variable model each tree is one observation, whereas in the taper model each tree represents multiple observations. This implicates a tendency toward more sensitivity when posing tests of hypotheses, basically because when exploring all along the stem there are more degrees of freedom in the error and the hypotheses are tested more successfully in the taper parameters. ^{Clutter (1980)} applies a total volume Schumacher-Hall-type model; when the total volume model is fit independently, the estimators differ, basically because in one the deviations on the stem are minimized and in the other the deviations on volume are minimized. With the values of these parameters, we were able to make useful total and commercial volume calculations on the stem.

Taper and compatible volume systems fit

SUR estimation decreased the standard error and the statistically significant level of the estimators increased, in both taper and commercial volume, and made the estimators more efficient (^{Rose and Lynch, 2001}). For this reason, when conducting the test of additionality in the form parameters β and δ, significant differences were found by addition to Tuxtepec (β₁≠β₂≠β₃≠β₄≠β₅≠β₆, p≤0.05; δ₁≠δ₂≠δ₃≠δ₄≠δ₅≠δ₆, p≤0.05). Thus, the effect of the dummy variable was significant in the systems (Table 5). This coincides with the findings of Quiñonez et al. (2014), who reported differences in the parameters that define taper and commercial volume for five pine species.

Table 5 Statistics of the ^{Fang and Bailey (1999)} systems for Spanish cedar progenies established in Tezonapa, Veracruz, Mexico. 

^†EE: standard error of parameters.

The ^{Fang and Bailey (1999)} model, case I-a, assumes that the β and δ form coefficients on both sides of the system are similar for each of the progenies (the form coefficient in delta calculated as ff=δ _i xk). However, Tuxtepec keeps the best form (β₀=0.45-δ₀=0.41). In contrast, Comala possesses the most conical stems (β₆=0.33-δ₆=0.32). The estimators are similar when compared with the first approach of the combined variable model and the direct form factor, but the system is more sensitive and detects differences in all the geographic sources by addition to Tuxtepec.

The effect of thinning to establish the seed orchard could have stimulated diametric growth of the trees by redistributing future growth among a few individuals. However, because thinning was too strong, it possibly promoted greater cone-like taper, low branches and therefore, thick nodes and less dense wood because diametric growth was too accelerated. For this reason, it is recommendable to maintain relative competition among trees (^{Rosso and Ninin, 1998}; ^{Chaves et al., 2013}). Moreover, the sources of origin of each provenance possibly had an influence in the geometry of the stems given that they are from an extensive geographic area and are adapted to environmental conditions different from the site where the trial was established. This is a common phenomenon called genotype-environment interaction and the reason that introducing provenances from the ecological zone most similar to the introduction site is advised (^{Chambel et al., 2005}).

Local growers, before using unknown sources of Spanish cedar germplasm, should consider the volumetric proportions that plantations with Tuxtepec progenitors could contribute. Nevertheless, it would be useful to conduct more in-depth investigation to study, for example, the degree to which the progenitors transmit their stem characteristics to their descendants.

The systems had high values in the adjusted coefficient of determination (R ²>0.94)., low RMSE, CV, and E (Table 6), and therefore, can be used to estimate a distribution of products for each provenance. Nevertheless, the ^{Fang and Bailey (1999)} system, case I-a, is recommended because the form factors have better behavior in both equations of the system. Moreover, all its regression coefficients were highly significant. The model selected with dummy variables is equivalent to having seven specific models for each group of progenies, which are statistically different.