Age Contributes to Volume Estimation and Form Factor of Pinus Pseudostrobus Lindley in Commercial Forest Plantations from Western Mexico

ABSTRACT Sectional equations and mathematical volume models are a reliable way to estimate carbon sequestration and storage, which is a key foundation for forest management and conservation. The objective of this study was to assess stem volume and form factor through the classical sectional method to then, using five regression models commonly used for forest management, identify the most suitable mathematical model to estimate the stem volume in a commercial forest plantation (CFP) of Pinus pseudostrobus Lindley in the Comunidad Indígena de Nuevo San Juan Parangaricutiro, Michoacán, in Western Mexico. By using 10, 15 and 20 yr.-old sampling points and two sampling methods (destructive and nondestructive), we found a form factor 0.42, 0.48 and 0.51 and stem volume of 0.098 m3, 0.400 m3 and 0.804 m3 for the three ages assessed, which presented diameter classes (DC), from 10 to 45 cm. The mathematical models identified that age of plantation determines stem volume and form factor, and the models that best fit volume estimation were the Schumacher-Hall model and the Australian model with an R2 adj range between 0.89 and 0.99. Estimation of stem volume is of vital importance to assess the income generated by the timber industry, and relevant for forest conservation, management, and carbon sequestration studies.


Introduction
For natural forests and plantation management, conservation plans or timber harvesting, obtaining the volume of tree trunks is important to estimate the amount of existing and usable timber in a forest. It is crucial in forest measurements to estimate growth and biomass based on stem volume, as well as in those studies that seek to explore the forest role in carbon sequestration and storage and ensure the sustainable use of natural resources (De Eugenio et al., 2018;Wegiel & Polowy, 2020).
There are different methods to quantify the stem volume, such as sectional methods, methods based on volume equations and those using the profile or taper functions, which is called as classical theory (Corral-Rivas et al., 2017;Dávila et al., 2012). The sectional methods use a geometric adjustment variable for the cylinder as the general reference solid. This variable is called as the form factor, which is a correction factor to consider the tree form and the taper along the stem (Coronel Toro, 2019; Dávila Molina, 2017). The sectional methods determine the form factor of a tree stem through mechanical theories (Alanís- Rodríguez et al., 2021). These methods assume that the main effect determining the form of a stem is the resistance to the wind and they predict a cubic paraboloid form (Prodan et al., 1997). Another group of theories assumes that the main effect is resistance to its own weight and predicts a neiloid form (Cruz de León & Cruz de León, 2006;Vásquez-Bautista et al., 2016). Over the past century, extensive work has been conducted in this field and Dean and Long (1986) retrieved the sectional method theories and presented them in a concise and simple way.
The current theory, using mathematical models for calculating the volume of wood in forests, is easier to adjust for different purposes such as timber production and wood conservation (Cancino, 2012) and it is easy-to-use tools that allow to obtain reliable volume values (Rodríguez-Ortiz et al., 2020). These mathematical models use diameter and volume, which present high correlation. However, the volume estimation will depend on the model used (Akossou et al., 2013;Cancino 2012). These tools have been used from the 1970s to the present day, being preferred by timberland companies managing natural forests and commercial forest plantations (CFP), increasing their use in the last 10 years for forest conservation and natural resource management (García et al., 2016;Hernández-Hernández -Ramos et al., 2018;Ramos et al., 2018).
Currently, there are many mathematical models of volume, both linear and non-linear (Heredia-Acuña et al., 2019;Lana et al., 2015;Prodan et al., 1997). All these models use the classical theory of sectional methods as the basis. For the development of these models, variables such as the normal diameter (ND), total height (H), diameters of ends or middle part of stem section (d) and height of section or log (hoL) are required (Cruz De León et al., 2016;Cruz De León & Uranga-Valencia, 2013;García-Espinoza et al., 2019). Newton's sectional volume method has been cataloged as the most accurate since it has been tested using the Archimedean principle of fluid displacement (Prodan et al., 1997;Riecke, 1849). However, its application is somewhat complex, as it needs to be applied individually, while a mathematical model of volume can be applied directly to any individual or forest stand for which model has been adapted, as well as it can provide a value close to the real one, it does not require a destructive method and its use is more economically efficient (Del Río et al., 2015;West, 2009).
Generally, methods to estimate volumes consider diameter classes but not age of the individuals (García et al., 2016). Some form factor studies, such as those developed in commercial forest plantations by Cruz de León and Cruz de León (2006); , reported significant differences in form factors among ages. They also found that a single diameter class may consist of different ages due to the presence of suppressed trees. Compared to those volume estimations, which only consider diameter, the methods considering age result in better quantity and quality of wood volume per stem, which influences forest wood volume and timber volume (West, 2009). Therefore, it was identified that in CFP, it is essential to consider the age of individuals when applying volume models in order to obtain better precision in the estimated values.
In this study, we used Pinus pseudostrobus Lindley, commonly called as smooth bark Mexican pine, as our model system to estimate the volume of wood in three different CFP's ages. P. pseudostrobus is considered as one of the forest species with the highest commercial value due to its phenotypic characteristics, good resin production and wood (2013; García et al., 2016;Hernández-Hernández-Ramos et al., 2018). In addition, P. pseudostrobus is one of the most used species in different silvicultural projects in Mexico due to their ecological association with other 25 and 34 species from the south to the center of Mexico, mainly of the genera Pinus, Quercus and Abies. Specifically, the Comunidad Indígena de Nuevo San Juan Parangaricutiro (CINSJP) located in western Mexico has developed forest management practices, and turned into a national and international benchmark due to the sustainability and determination of the community to stand out as a timberland company. The CINSJP own natural pine and oak forests, as well as plantations of several pine species, especially P. pseudostrobus (Heredia-Acuña et al., 2019).
This study aimed to assess the stem volume and form factor in trees of P. pseudotrobus Lindley for three ages: 10, 15 and 20 yr.-old, in CFP in the CINSJP through the classical sectional method to then, using five regression models commonly used for forest management, identify the most suitable mathematical model to estimate the stem volume in a commercial forest plantation (CFP). To do this, we used two types of sampling methods, destructive and nondestructive, which initial volume by sectional methods (classical theory) and a natural form factor (FF 0.1H ) were calculated for each age, to later identify the most suitable volume model among five mathematical models: Schumacher-Hall, Schumacher-Hall linearized, Spurr -Combined variable, Australian and Thornber. We selected the best models using a weighting table, mainly a higher adjusted coefficient of determination R 2 (R 2 adj ) and lower coefficient of variation (CV). We specifically asked: 1) the stem volume stored and form factor of P. pseudostrobus Lindley in the different sampling points vary according to age? 2) Are there significant differences between destructive and nondestructive sampling to calculate stem volume and form factor? and 3) Is there a general mathematical model for calculating volume for the entire plantation or a specific model for each age of the plantation. We hypothesize that, individuals within the same diameter class (DC), will have different ranges of volume (V) values due to the age in which they are (Talita et al., 2018). Then, we expect that destructive sampling will present better precision than nondestructive sampling; therefore, there will be statistically significant differences between them (Vásquez-Bautista et al., 2016). Finally, we hypothesized that age-specific mathematical volume models are expected to present a better value of estimated V, than an Age General Model (AGM; Blanco-Flórez et al., 2014;Prodan et al., 1997). Our study shows that obtaining volume values of P. pseudostrobus wood in a more confinable, precise and fast way, improves the productive performance in CFPs, which can be applied for forest management, use and conservation plans, as volume is an essential variable for estimating biomass, carbon sequestration, and storage contributing to enhancing greenhouse effect and climate change.

Study area
The present study was conducted in commercial forest plantations (CFP) of the Comunidad Indígena de Nuevo San Juan Parangaricutiro (CINSJP), which has an area of 18,138 ha and it is in the Western of Mexico in the state of Michoacán, 00"-102° 17' W. The climate is temperate humid type C (w2), and the average annual temperature of the sites where P. pseudostrobus is present is 15.2°C, with an average annual rainfall of 1,125 mm (Sáenz-Romero et al., 2012). The soils are of volcanic origin and the units of Andosol, Regosol and Feozems. The type of dominant forest cover corresponds to a natural association of pine-oak forests, which are under forest management, with the Mexican Method for the Management of Irregular Forests (MMOBI) and the Silvicultural Development Method (MDS); the harvest of annual wood is approximately 65,000 m 3 (Velázquez et al., 2015). The most representative tree species are P. pseudostrobus Lindl., Pinus devoniana Lindl., P. montezumae Lamb., P.  (García et al., 2016). Since 1983 the CINSJP has conducted the forest management to obtain sawed wood products such as boards, planks and rollers, for construction and for fine finish products, such as furniture, staves, moldings, floors and constructions. Forest industry, including logging, sawmill, reforestation, furniture production and resin extraction, is the most important economic activity with more than 35% of inhabitants being related to this industry (García-Espinoza et al., 2019).

Study species
Pinus pseudostrobus Lindl. (Pinaceae) is widely distributed throughout highlands from 1600 to 3200 masl in Mexico and Central America in an extensive geographic range (it has been recorded in 20 of the 32 states of Mexico). In the study area, reproductive individuals can reach up to 40 m in height with very straight stems and up to 100 cm in stem diameter (Morales-Arias et al., 2018;Rodríguez-Ortiz et al., 2020). P. pseudostrobus may be found forming pure forest stands or pine-oak mixed forests. Its timber is one of the best qualities for construction and production of pulp for paper, plywood, fine furniture, handcrafts, and cabinetmaking. Moreover, it is one of the species with the highest production of resin (Rodríguez-Ortiz et al., 2020). P. pseudostrobus is one of the most important timber species in Mexico, and especially in Michoacán. Specifically, the CINJSP uses the species, as the main raw material, for the primary and secondary production of timber products (Velázquez et al., 2015).

Experimental design
We did an exploratory sampling using 10-8 × 20 m transects established randomly at six sampling sites of the commercial forest plantation of the CINSJP. Each sampling site refers to the age of the plantation: 5, 10, 15, 20, 25 and 30 yr. At each sampling site, we measured the normal diameter (ND: diameter of the stem measured at 1.30 m; (Prodan et al., 1997) and total Height (H) of all trees within the transect in order to identify the sample size, which must be at least 30 trees for sampling (Urbano et al., 2018), and diameter classes available (Murrieta et al., 2007). Sampling size and diameter classes were useful to identify the most suitable sampling site to perform this experiment. Therefore, three sampling sites of P. pseudostrobus of different ages were selected: Tazaman (10 yr. old), El Tejamanil 1 (15 yr. old) and El Tejamanil 2 (20 yr. old) where we additionally estimated stem volume (Eq. 1) and form factor (Eq. 3) of all trees using two different sampling methods: 1) Destructive and 2) nondestructive described below.

Sectional methods to estimate volume and form factor
(1) Destructive sampling method. Within the Commercial Forest Plantations (CFP) of CINSJP, we selected 60 trees of 10-45 cm of normal diameter at each sampling site, having commercial features, including straight stems, tall branches and good health status. For the selected trees, we measured basal diameter (BD), normal diameter (ND) and crown diameter (CD) in two directions, commercial height (H c ) and total stem length (H; Urbano et al., 2018). Later, we felled the selected 60 trees to perform the experiments.
(2) Nondestructive sampling method. This methodology requires the use of an electronic Criterion dendrometer (RD1000). We chose 10 trees for each diameter class (DC) of the selected sampling sites, for a total of 120 P. pseudostrobus trees. First, we measured the tree stems from the bottom to the top, following the destructive sampling method, dividing the total height by ten and measuring the diameters with bark of each section. Then, we measured the normal diameter, commercial height, total height and diameter of the crown (Prodan et al., 1997;. For both the destructive and nondestructive methods, the truncated cone sectional method equation (equation 1) was used to obtain the volume of wood: where V Tc = Truncated cone wood volume. L = Length of the stem or log.
S 0 and S 1 = Areas of the stem's outer cross-sections (major and minor), respectively.
For the form factor, we used the coefficient between the real volume (V r ) of the tree and the volume of a geometric reference body (V s ; equation 2; (Cancino, 2012)., whose dimensions correspond to the general dimensions of the tree (ND and height). The volume of the reference solid is also called apparent volume (West, 2009).
where FF = form factor. V r = real volume of the stem.
V s = volume of the reference solid.
When the reference solid is a cylinder V s = g * h, where g = basal area and h = height.
For the form factor at 10% of total stem height, in the Equation 3, the influence of the tree size on the form factor is eliminated by measuring the diameter for the reference solid at a percentage of the tree's total height. Thus, the diameter is obtained at the same relative height in trees of different heights. The form factor corresponding to the diameter measured at the height of 10% of the total height and using the Huber method to estimate sectional volumes is called the natural or Hohenadl form factor (equation 3). Therefore, the natural form factor was obtained from the following relationship (note that the volume of the cylinder in this case is Vs = g 0.1 H; Prodan et al., 1997).
where FF 0.1H = form factor at 10% of total stem height. g 0.1 = stem cross-sectional area at one-tenth of total height, H = total height of the tree.
We tested for normality, using Shapiro-Wilk and Kolmogorov-Smirnov test according to the sample size, and homoscedasticity using Levene test. Then, we performed an ANOVA using the software SAS. 9.4 TS to test differences in volume and form factor comparing: i) the destructive and nondestructive sampling methods and ii) ages of the CFP.

Mathematical forest models to estimate stem volume
Based on the values already calculated using the sectional method for stem volume, height and normal diameter, five mathematical models (Table 1) commonly used in forest sciences were used to estimate the stem volume. These models are commonly applied in the construction of volume tables (Clutter et al., 1983;Prodan et al., 1997;Santos et al., 2019). According to the fulfillment of assumptions, the best model for calculating the volume of the three evaluated ages of the CFP was identified. In addition, we identified the correlation between the volume estimated by each model and the real volume (obtained from the sectional method). To select the best model, seven different statistical parameters related to precision and volume variation of each model were ranked, giving a weight 1 to the lowest value and increasing weights for the other models according to Hernández-Hernández- Ramos et al. (2018), except for the R 2 adj that a high value is given a weight 1. The statistical parameters evaluated in the weighting table were calculated F value, variance (Var), calculated standard error (SEE), mean square error (RMSE), coefficient of variation (CV%), adjusted coefficient of determination (R 2 adj ) and values of the Shapiro Wilk tests for ages 10 and 15 and Kolmogorov Smirnov for age 20 (SW/KS).
Subsequently, the sum of the weights resulted in a weighted total score (WT) and the best model for each age were those with the lowest WT (Ramos et al., 2013). Finally, a general model was developed lumping individuals of all ages (Age General Model, AGM) for each of the five models. The models that were optimal for each age, we checked the homoscedasticity for each model, considering that the errors or residuals (y-ŷ) of the regression must be distributed with equal variance (σ2) in all range of values of the independent variables (Box-George et al., 2015;Tlaxcala-Méndez et al., 2016). If the assumption is not fulfilled, then the model would have a heteroscedastic behavior and therefore it cannot be accepted as a final model. We tested for homoscedasticity of the residuals using the Levenne test. To develop and test these models we also used SAS. 9.4 TS software.

Results
We found that Pinus pseudostrobus in CISJNP showed an important variation in diameter for trees of specific ages, with older trees being more variable in diameter. Thus, for the age 10, there were two diameter classes, while for the age 20, up to six diameter classes were found (range: 15-20 and 40-45 cm; Table 2). Height also presented variation, even within the same age and especially for older ages. For age 20, we found a mean height of 17.2 m (SE ± 0.652) for the smaller size class (ND 10-15 cm) and 25.7 m (SE ± 0.750) for the larger class (ND 40-45 cm). Considering both sampling methods (destructive and nondestructive), we found a big variation in volume with a mean of 0.532 m 3 (± 0.08) with minimum and maximum values of 0.060 to 1.356 m3, respectively (Table 2). Table 2. Volume and form factor for Pinus pseudostrobus of different age and by applying a destructive and nondestructive method. DC = diametric class; ND = normal diameter (cm); H = height (m); VD = volume by destructive method (m3); VND = volume by nondestructive method (m3); FFD = form factor by destructive method; FFND = form factor by nondestructive method; SE = standard error. Our results showed a smaller variation for the form factor (FF 0.1H ), with values from 0.457 to 0.564 with a mean of 0.507 (SE ± 0.032). The observed variation in V and FF 0.1H was given mainly due to age and diameter class with larger volume and form factor for older ages (Table 2). It is also important to highlight that those trees in the same age, but different size class resulted in very different volume estimations due to the influence of height and diameter (Table 2). In general, both the destructive and nondestructive methods attained similar volume estimations and no statistical differences were found (Table S1). The volume based on the sectional methods was used to yield a volume of 0.534 m 3 (SE ± 0.02) for the destructive and 0.532 m 3 (SE ± 0.02) for the nondestructive method ( Table 2). The same pattern was found for the form factor with an estimation of 0.506 (SE ± 0.011) and 0.507 (SE ± 0.011) for destructive and nondestructive, respectively (Table 2). We did not find significant differences among the destructive and nondestructive method either for form factor or volume for none of the ages (Table S1).

Mathematical forest models
In general, lumping destructive and nondestructive methods and aiming on age, we found a clear differentiation among ages in the three dendrometric variables (Table S2). The mean (SE) for ND at age 10 was 14.4 cm (± 0.55) and for 20 yr. was two-fold larger. Height also had a similar trend going from 10.06 m (± 0.22) in 10 yr. to 20.62 m (± 0.36) in 20 yr. For volume, we found a mean (SE) of 0.098 m 3 (± 0.01) for 10 yr. and for 20 yr. the volume increased 8.9-fold (Table S2).
We found that all five mathematical forest models for all ages predicted successfully the volume (V) based on normal diameter (ND) and height (H), with R 2 adj significant values going from low (R 2 adj = 0.8822) to very high (R 2 adj = 0.9902; Table 3). The weighting table is shown below (Table 3). Table 3. Results obtained for each model in the 3 assessed ages and an age general model (lumping the three ages) for Forest Commercial Plantations of Pinus pseudostrobus at CINSJP at Western Mexico. Statistics are as follow: F = test statistic; Var = variance; SEE = standard error of estimation; RMSE = root mean square error; CV% = coefficient of variation; R2 = R square; R2adj = R square adjusted; SW/KS = Shapiro Wilk and Kolmogorov Smirnov. Numbers in brackets indicate the weight to identify the best performed model for each statistic and the WT represent the weighted total, which is the sum of all weight. The best model for each age were those with the lowest WT indicate in bold according to (Ramos et al., 2013  Based on the Weighted Total, which consider the weight of all statistics parameters, we identified the best models. However, none of the five models reached the highest WT for all ages or AGM and each age had a specific best performed model. Thus, for 10 yr. and 15 yr., the best model was Schumacher -Hall, while for 20 yearsr and the general model (AGM), the Australian was the best performed model (Table 3;  Table 4). These two models included in their equation three and four regression parameters, respectively, and specifically, the Australian include a combined variable (ND*H; Table 4).
All the best performed models for each age showed their residuals distributed with equal variance along the values of the axes ND, H and the combined variable (ND*H; Fig. S1) and we found p-values ranging between 0.825 and 0.957 based on the Levene test, indicating normality of residuals (Table S3). Below are the graphs of the mathematical models chosen for each age and the AGM model, in Figure 2.

Discussion
Volume variation occurs since the growth and allometry of a tree depend on factors such as the position of the stem in the forest, the availability of nutrients, the state of health of the individual and the microenvironment where the tree is established (light, water availability, etc. (Cancino, 2012;Tlaxcala-Méndez et al., 2016;Veintimilla et al., 2019). These events are the main cause of the large variation in estimated volume (V) and form factor (FF). In our study, we found a large variation in size, both in diameter and height, within the same age, and especially for individuals aged 20 yr., we found up to six diametric classes, in trees with a range of 16.9-44.9 cm of normal diameter and a height of 17.9-28.3 m. Although a large variation was found within the same age, we emphasized the statistical differences in both FF and V between plantations of different ages. These differences have serious consequences for the forest industry, as it will directly affect the FF and V estimation and consequently, the projected wood production and quality. In general, for P. pseudostrobus in the forestry industry, a fixed (0.45) or sometimes arbitrary (0.55-0.70) standard form factor is used (García et al., 2016). We highlight the correct estimation and use of the appropriate FF for the benefit of plantation owners and forest management, as well as the use of age-specific FF as these changes in allometric relationships result in erroneous V estimates (Phiri et al., 2016).
Commercial forest plantations (CFP) must have a rigorous forest management plan to meet high-quality standards in their final products. Pruning, fertilization, and thinning must be carried out in the time and manner established in the management plan Subiakto et al., 2016) to avoid the presence of low diametric classes or trees that do not correspond to the age of the plantation (Orozco, 2008). Appropriate forest management activities carried out in CFP generally guarantee better estimates of the FF compared to those of natural forests (Tlaxcala-Méndez et al., 2016;Uranga-Valencia et al., 2015). In this study, we estimated a mean FF value of 0.506 (SE ± 0.01), higher or similar when compared to other studies such as those of Uranga-Valencia et al. (2015), who obtained an FF between 0.447 and 0.506 for Pinus patula in three different regions of Mexico. Several other studies with Pinus species in other countries (Dávila Molina, 2017;Jacobs et al., 2019;Phiri et al., 2016) have found values between 0.47 and 0.57. In the previous research in the same study area in the CFP of CINSJP in Michoacán, but in natural forest, García-Espinoza et al. (2019) found an average FF of 0.46 (SE ± 0.03), showing that the managed forest presents better parameters for V estimations, such as FF database on our results show that CINSJP can establish objectives for forest conservation and management in natural forests, and for improving the forest quality, through constant monitoring by using forest surveys.
Forest surveys serve as a monitoring tool to observe the state and quantity of stem volume in natural forests and CFPs (Cancino, 2012). In cases where there are no V estimation models specific to the sampling area, destructive sampling is required in order to have a real V calculation. To do this, a sectional method is performed, which is generally not feasible, since standing trees are the main product of the forest management program for future extraction (Prodan et al., 1997). Therefore, nondestructive methods emerge as the best option for faster V estimation as it is not necessary to cut down trees, although accurate estimates are required. Our findings show that the estimates of P. pseudostrobus V and FF based on a nondestructive method (mean V SE: 0.532 m 3 ± 0.026; mean FF 0.1H SE: 0.507 ± 0.013) resulted in estimates like those obtained from the classical theory that uses destructive methods (mean V SE: 0.534 m 3 ± 0.027; mean FF 0.1H SE: 0.506 ± 0.011; Table 2 and Table  S1). Therefore, we highly recommend, based on our findings, the use of the nondestructive method to estimate V and FF in CFP of P. pseudostrobus in order to guarantee forest and natural resources sustainability that CPF provides.
The absence of statistically significant differences between destructive and nondestructive methods opens up many possibilities for factors that may determine variability in V and FF among DCs. Therefore, based on our results, we confirm that age is a very important factor to obtain accurate and reliable results close to the actual V values. This statement agrees with Talita et al. (2018) regarding CFPs oriented to wood production. We also found that the Schumacher-Hall model fit better at 10 and 15 yearr, while the Australian, which includes the combined variable: diameter and height (ND*H), was a better model for 20 years and for the AGM. This indicated that each model better adjusted the ratio of ND and H, and this may be related to changes in growth and allometric relationships throughout the life cycle of the species (Akossou et al., 2013). Meeting criteria for better model fit is of vital importance when deciding the suitability of the models, as failing, the V estimate will be deficient and the predictions will be inaccurate (Álvarez-gonzález & Rojo-alboreca, 2007;García et al., 2016;García-Espinoza et al., 2019;Martínez-Angel et al., 2019;Uranga-Valencia et al., 2015). For the Age General Model (AGM), which included individuals of all ages, we found the Australian as the best model to predict V, being recommended in the literature for natural forests, when the age of a plantation is unknown, dendrochronological studies are difficult to complete (Hernández-Hernández- Ramos et al., 2018) or for academic and training purposes, since it is not necessary to fell the tree. However, it should be considered that the accuracy of this model in general is lower compared to models for ages 10, 15 and 20 yr. Therefore, our results show that it is important to partition into ages and diametrical classes when estimating V and FF for future research in CFP and in natural forests where dendrochronological analyses are possible to perform.
Methods to estimate V and FF in both planted and natural settings are key activities for forest management, conservation, and sustainable use. We assessed mathematical models for V estimation, for each specific age, with good fit and precision, suggesting the limitation that when performing volume estimation of P. pseudostrobus CPF, an accurate age estimation of the plantation and a precise knowledge of the mathematical model to use should be taken into account. When making a wrong decision in one of these two aspects, there will be a high probability that the field technician fails to estimate correctly the stem volume (Vásquez-Bautista et al., 2016;West, 2009). The advantages of having specific mathematical models for each age are that they allow monitoring surveys to be carried out at each planting site, without the need for destructive sampling, since these mathematical models offer estimates of V very close to the real ones. Therefore, not requiring destructive sampling to have optimal values of V allows the appropriate management and sustainability of forests, and consequently, the estimation of biomass and carbon sequestration rates, which allows to monitor the amounts of carbon stored in both planted and natural forests and discriminating according to planting ages, diameter classes, or other variables of interest in forest management and conservation.

Conclusions
The models for each age presented a better fit than the AGM, with more precise volume estimates, which allows us to recommend the use of volume estimation models for each age, in ranges of 5 years in CFPs of P. pseudostrobus. We also recommend the use of specific form factor by age range as we showed in this study, as it may help to improve not only the estimation of the stem volume, but also to control the yields in wood production and use for each plantation improving forest management strategies. P. pseudostrobus is one of the most important forest resources in Mexico. Specifically, for the CINJSP, it represents the best species for timber and resin production from which the community makes its most important profits. We believe our study can contribute to a quick and precise estimation of volume and form factor and in general to guide the community for better forest management practices. We also suggest following rigorously the forest management plan, especially in relation to clearing, pruning, and thinning to obtain better volume and better profitability and sustainability of the forest.