Artificial intelligence tools in predicting the volume of trees within a forest stand

The goal of this study was to train, validate, select and evaluate artificial neural networks (ANN) to predict the individual volume of wood in eucalyptus stand, based on the diameter at breast height (DBH) and DBH with the total height (Ht). Data was obtained from a plantation of Eucalyptus urophylla ST Blake of seven years of age, located in the state of Goiás, Brazil. Sixteen plots were randomly set in this area, from which the variables diameter, total height and volume were accounted. The volume of all the trees in each plot was measured by the Smalian method; afterwards, the data were checked for normality using the Shapiro-Wilk test. Sequentially, perceptron network settings (ANN1 = DBH and Ht; and ANN2 = DBH) were trained using sigmoid activation functions and resilient propagation (Rprop) algorithm. In addition, a root-mean-square error (RMSE) of less than 1% was adopted as stopping criterion or when this rose again. The selected ANNs presented low variation among the task-specific training indices, selection and evaluation, showing correlation ( ̂) between predicted and observed volume (0.9945 and 0.9898), and RMSE from 1.75 and 2.22%, respectively. The Shapiro-Wilk test highlighted non-normality of data distribution; hence, various selected ANNs were subjected to the Kruskal-Wallis test for validation, as well as for comparison with each other and sequentially submitted to the overall group difference test. The test demonstrated that both ANNs were able to predict tree volume; leading to the conclusion that multilayer perceptron neural networks (MLPNNs), using just one neuron inputthe diameter, are as precise and accurate as networks using two neuronsthe diameter and height, in order to predict individual volume of E. urophylla.


INTRODUCTION
Planted forests cover about 264 million hectares, comprising nearly 7% of total forest area, with the largest part located in China, India and the United States (61%).Brazil has 7.6 million hectares of planted forest (3%) and contributes to 17% of all wood harvested each year.This contribution arises from the high productivity of *Corresponding author.E-mail: Isaltinoab@gmail.com.
Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License plantations, mainly of Eucalyptus trees, which represent 72% of the planted forest area planted in the country (IBÁ, 2014).
It is estimated that among all planted trees in the world, Eucalyptus makes up nearly 38% of that (Pérez-Cruzado et al., 2011).From that, India has the highest planted area, with 22% of this tree genus.However, Brazil is the second place (21%); and new plantations have been growing in the country due to wood demands from the industries that use it as raw material.This growth comes also from the need for restoration of native areas, which minimizes environmental impacts.
Besides being the main tree genus grown in Brazil, Eucalyptus is the most planted tree in the tropics (Epron et al., 2013) because of its rapid growth, yield, easy adaptability, species diversity and wide range of wood use.However, before being marketed, plantation production measurements have to be held, particularly the wood volume (Miguel et al., 2014).
Estimating the volume of forest stands derives from quantitative inventories.This operation consists of measuring representative samples that are called plots (Binoti et al., 2013).Subsequently, this volume is rigorously correlated with easy-to-measure variables, such as diameter at 1.30 m above soil (diameter at breast height -DBH) and total height (Ht).Then, regression techniques and adjustments of volumetric equations are adopted as a routine procedure.
Several volumetric models can be applied to these measures, which are adjusted linearly or non-linearly, being divided into single input models (DBH) and dual input (DBH and Ht) besides their multiple combinations.Nevertheless, regarding statistic methods used to measure the quality and accuracy of the adjustment, the dual-input models are preferred because they are more accurate, as stated by Thomas et al. (2006), Azevedo et al. (2011) and Miguel et al. (2014).In addition, this type of modeling requires statistical assumptions such as data normality or linearity (Egrioglu et al., 2014).
One option for exempting these statistical assumptions to represent nonlinear relationships between predictor and predicted variables is the use of artificial intelligence (AI) techniques and resources.The use of these tools in growth and production modeling is still new and unexplored.However, efforts have been made and with promising results (Diamantopoulou, 2005;Gorgens et al., 2009;Castro et al., 2013).Among these techniques, artificial neural networks (ANN) has gained prominent position (Silva et al., 2009;Binoti et al., 2013;Miguel et al., 2015).Forest stand modeling through ANNs allows greater accuracy in production estimates, as well as being a good aid at making decisions (Castellanos et al., 2007).
ANNs are modern data processing systems whose design, structure and operating principles are based on biological neural system, interconnected to perform a particular task.It has as fundamental element an artificial neuron, which receives as input operating parameter values and returns an expected result as output.
According to Wang et al. (2010), artificial neuron is a simplified model and related real neuron, whose basic properties are the suitability and reproduction of information based on connections, which is the information processing unit of a neural network.The networks has as basic characteristics adaptive learning, self-organizing capacity, robust structure distributed in parallel (layers), efficiency in learning and generalization, besides being tolerant to outliers, are able to model different variables and their nonlinear relationships, as well as enabling quantitative and qualitative variable modeling (Haykin, 2001;Kuvendziev et al., 2014).
This work aimed to train, validate, select and evaluate artificial neural networks (ANN) to predict individual wood volume in a Eucalyptus stand, using diameter at breast height with total height (RNA1) and only diameter at breast height (RNA2).

MATERIALS AND METHODS
Data were collected from a plantation of Eucalyptus urophylla S. T. Blake of seven years old, planted at spacing of 3 x 3 m (1,111 trees per hectare).The area has 110 hectares and belongs to the Cooperativa Agroindustrial dos Produtores Rurais do Sudoeste Goiano (COMIGO) (cooperative society), in the city of Rio Verde, southwest of the state of Goiás, Brazil.The area lies at an average altitude of 700 m, limited between 18° 00' 45'' to 18° 01' 45'' south latitude and from 50° 52' 45'' to 50° 53' 15'' west longitude (Figure 1).According to Köppen, local climate is humid tropical (Aw type) with two distinct seasons: a dry one (fall and winter) and another wet with pouring rains (spring and summer).Each year, rainfall ranges from 1,200 to 1,500 mm, averaging around 1,300 mm, and average temperatures between 20 and 25°C (Siqueira-Neto et al., 2011).

Sampling, variables and data analysis
The authors carried out a pre-exploratory forest inventory within the study area.Fifteen 400 m 2 plots were delimited with the aid of a measuring tape.For evaluations, a fixed area method with a simple random sampling process was adopted (Husch et al., 1993).
After the plots were set, diameter, total height (Ht) and volume were measured.For diameter, a diameter tape performing measurements at 1.3 m above the ground, which is called diameter at breast height (DBH) was used.Then, all standing living timbers within the 15 plots (605 trees) were cut, the total heights were obtained directly and their wood volume was strictly measured.
Next, the distribution of trees was assessed by diameter class within a range of 2.5 cm each class.Scolforo and Thiersch (2004) mentioned that forest stands should be evaluated at a class range between 2 and 5 cm.Such evaluation is intended to represent a horizontal stand structure, while ensuring that all trees are sampled by their diameter class during the process of wood volume measurement, increasing the prediction consistency and accuracy.
Subsequently, wood volumes were measured by the analytical method of Smalian.According to Machado and Filho (2006), this method is widely used in Brazil, mainly due to its practical advantage and precision.Diameter was measured throughout the entire timber at spots previously set, disregarding the height of stumps with 0.1 m.At stem basal portion, section lengths were 0.2 m, while the remainder sections were 1.0 m.Therefore, diameters were measured at 0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 2.0 (0.7-m section), 3.0, 5.0 and 4.0 m, successively up to a minimum diameter of 2.0 cm.From this diameter until the top of the tree, the tips were accounted for.
The volume of each section was calculated by the formula of Smalian, yet the tip volume was obtained using the formula for calculation of cone volumes, as described in Machado and Filho (2006).Shortly after measuring the volumes and grouping into diameter ranges, trees were randomly selected to compose the database for training on the ANNs, within each class.The optimal number of timbers in total and by class was given by the following expression: Where, "n"= optimal number of timbers in total and by class; "t"= studentized value; CV%= coefficient of variation; E = permissible error (5%) for "α" of 0.05.
As usual, diameter class distribution in forest stands of the same age tends to normality.Therefore, by statistical means, the following measures were assessed: central tendency measures, variance, standard deviation, coefficient of variation, skewness and kurtosis.The coefficient of variation (CV), as being dimensionless and enabling comparison of variability with other variables, consists of an interesting measure, being its values classified according to the criteria proposed by Gomes (2000), as follows: low (CV < 10%); medium (10% < CV < 20%); high (20% < CV < 30%); and very high (CV > 30%).Asymmetry indicates the trend towards greater concentration of data within a central point, which in turn was examined regarding the following aspects: Symmetric: mode = median = arithmetic mean.Asymmetry to the right or positive: if mode <median <arithmetic mean.Asymmetry to the left or negative: if mode > median > arithmetic mean.
If Pearson's skewness coefficient, in modulus, was between 0.15 and 1.00, asymmetry would be considered moderate.If it was higher than 1.00, asymmetry would be considered strong.Kurtosis refers to the degree of flatness or peakedness of the distribution, which is usually considered in a normal theoretical distribution.It occurs due to concentration of values near the average.Pereira and Tanaka (1990) established three curve types regarding the kurtosis percentile coefficient: Leptokurtic: distribution having a relatively high peak with a negative excess, or kurtosis coefficient <0.263.Platykurtic: curve has a flatter peak with positive excess, that is, kurtosis coefficient > 0.263.Mesokurtic: intermediate curve, with kurtosis coefficient of 0.263.Numerical variables were linearly normalized within a range of 0 to 1, for training of the ANNs (Heaton, 2011).The input layer was formed either by two (2) quantitative neurons (DBH and Ht), called ANN1, or one (1) neuron (DAP), called ANN2, in function of the response/output variable (volume).
An artificial neuron is the information processing unit of an ANN, being composed of "n" inputs x1, x2, ... xn (dendrites) and only one output y (axon).The inputs are multiplied by some k w parameters called weights (w1, w2, ... wn), representing the synapses.These values can be negative or positive.Currently, a basic model of artificial neuron can be represented mathematically as: Where, Yk = output of the artificial neuron;  = Activation function; Vk = result of the linear combiner, that is: . (3) The networks were also comprised of a single hidden layer architecture.According to Esquerre (2002), most of the time, the networks require a single hidden layer to solve nonlinearly separable problems.The number of neurons in this layer has been optimized by the Intelligent Problem Solver (IPS) tool of the Statistica 7 software (Statsoft, 2007); and as activation function, the sigmoidal was used.Sigmoid activation function is quite common in developing artificial neural networks, and in addition to a well-built network architecture, it can bring closer any continuous function with great precision (Ismailov, 2014); this function is mathematically expressed by: Where,  = sigmoid activation function; β = parameter determining the sigmoidal function slope; u = function activation potential.
The resilient propagation was used as training algorithm (Riedmiller and Braun,1993), and the training parameters were learning rate (μ) of 0.2 and momentum (η) of 0.9 (Gorgens et al., 2009).
First, all network weights were randomly generated (Heaton, 2011).Then, individual update value evolved during the learning process, based on the error function.Network training continued until error rate was reduced to an acceptable margin between the predicted and actual values, known as delta rule, or until a maximum number of times or cycles (Shiblee et al., 2010).
To estimate the individual total volume, 100-perceptron multilayer networks were trained, which are commonly known as multilayer perceptron (MLP).One hundred networks were trained using diameter and total height in the input layer, and other 100 networks using only diameter, totaling 200-trained networks.Trainings were carried by a supervised method, in which input and output variables were indicated for the networks.This feedforward type method uses unidirectional data flow algorithm without cycles (Haykin, 2001).
Several methods determine the time at which the training of a neural network should be terminated.According to Chen et al. (2014), excessive number of cycles can lead to a network overfitting; on the other hand, it is underfitted when it has few cycles, what impairs a maximum performance.However, to eliminate these problems, the mean square error less than 1% was used, or when the root mean square error (RMSE) increased again as a stopping criterion, as suggested by Chen et al. (2014).Thus, training was terminated when one of the criteria was reached.
Based on the correlation between the volumes observed and the estimated networks ( ̂), the best two ANNs were selected.The authors also took as a basis stability of the training indexes of the networks that were provided by the software during training, selection and evaluation phases.In these phases, these indexes should be stable, that is, there must be variation among such indexes, the root mean square error (RMSE) in percentage and the graphical analysis of residues.
The root-mean-square error evaluates the mean square difference between observed and estimated values.The lower the RMSE, the better the average accuracy of estimates, being optimal when it is equal to zero (Mehtätalo et al., 2006).
Where, ̅ is the average of total volumes originating from wood volume measuring; ̂ is the individual total volume estimated by the ANN; is the individual total volume from the wood measuring, and "n" is the total number of observations.
Once the two best ANNs are defined, if non-used sampled timbers presented regular destruction in accordance with the Shapiro-Wilk test (1965), they would undergo variance analysis (ANOVA); if not, they would undergo the Kruskal-Wallis' test (1952).Sequentially, they would undergo the aggregate difference test (Da%) for training validation, as well as comparison between the ANNs with one (1) or with two (2) neurons in the input layer (DBH and DBH/ Ht, respectively).
The software used were SISVAR and Action 2.7, which performed descriptive statistics and normality test.Yet the graphics were made through Microsoft Office Excel 2013, and neural networks were trained in Statistica 7 (Statsoft, 2007).

RESULTS AND DISCUSSION
The structure of a forest is mostly defined by size and distribution of trees per area unit.In this aspect, the diameter is the most important variable, being used for modeling, volume measurement and in the understanding of the forest stand growth.
Diameters ranged from a minimum of 5 cm to and maximum of 21.75 cm, and mean, median and mode of 15.47, 15.05 and 15.73, respectively.These values indicate that the diameters presented a negative moderate asymmetric distribution, as mean <median <fashion, the negative skewness is also justified by the asymmetry coefficient of -0.233.
The Shapiro-Wilk (1965) normality test presented pvalue of 1.97e -12 , indicating a non-normal distribution, as shown in Figure 2A, since data are not distributed on the line.Diameter distribution had class range of 2.5 cm, platykurtic type (Figure 2B) with kurtosis of 0.280 and variation coefficient of 19.37%, ensuring a mean variability (Gomes, 2000) and standard deviation of 2.92 cm.
In studies in Brazilian, forest stands as Eucalyptus urograndis (Miguel et al., 2014), Acacia mearnsii (Sanquetta et al., 2014) and Pinus taeda (Téo et al., 2012) were found to have descriptive characteristics similar to this study.This behavior is expected in stands of the same age in Brazil; however, it may differ according to species genetic improvement degree, silvicultural treatment and such a way as to raise or lower the frequency of trees management type adopted, in  with larger diameters, shifting the mode to the right or left of the average.Of the sampled and measured trees in the field (605), sampling intensity by diameter class (α = 0.05; 5%) had a total optimal number of 131 trees (28%) for training the ANNs.These trees were distributed into the different classes of diameter and height, as shown in Table 1.The remaining 471 trees composed of another independent database used for validation.
Afterwards, based the 131 sampled trees for training, two networks were selected.Both with an input layer, a hidden layer and an output layer.The input layer differed in the number of neurons, sometimes two (2), the diameter and height (ANN1), other times one (1) and only the diameter (ANN2).The use of a single hidden layer is supported by the "universal approximation theorem", which states that only one hidden layer is enough for a MLP network to do the approximation of any continuous function (Cybenko, 1989).Yet, the number of neurons was determined by the Intelligent Problem Solver tool (IPS) (STATSOFT, 2007), and presented different numbers according to the amount of neurons in the input layer.Therefore, in total volume estimate by DBH and Ht (ANN1), the hidden layer had five neurons (Figure 3A), but when it was estimated based on DBH only (ANN2), the number of neurons increased to seven in this layer (Figure 3B).This difference is attributed to a greater or lesser difficulty of the network in predicting the volume using one neuron (DBH) or two (DBH and Ht) in the input layer.Within this structure, the input layer is where the standards are displayed to the network (DBH and Ht); the intermediate layers (also called hidden or secret layer) are responsible for much of the processing, which may be considered as extractors of characteristics; and the output layer is one where the result is displayed (volume).

ANN training
Adjustment and accuracy statistics of both selected networks to predict individual wood volume of E. urophylla trees were satisfactory.The selected ANNs showed low variation between the levels of training, selection and evaluation, which consist of ideal results that show training stability (Binoti et al., 2013).The correlation between observed and predict volumes ( ̂) was 0.9945 and 0.9898, with root-mean-square error in percentage (RMSE%) between 1.75 and 2.22%, respectively (Table 2).Thus, the use of the ANNs has effectively estimated tree volume and at the same time exempts from basic assumptions of the standard mathematical modelling, such as normality and linearity of the forest attributes (Egrioglu et al., 2014).These attributes often undergo different mathematical transformations to be modeled in a traditional way, which may cause losses in quality and selection of models.
Artificial intelligence has great potential in diverse applications, especially in the areas of engineering and agriculture.However, for its application feasibility, Cartwright (2008) stated that there should be a direct relationship between the input parameters and the target responses.Overall, the networks are developed for nonlinear mapping from a set of inputs and outputs that are interrelated.In such cases, the ANNs are developed aiming to achieve a typical performance of a biological system, based on the interconnections of the elements, similarly to what occurs with biological neurons (Gürüler et al., 2015).In addition to these characteristics, ANNs had some advantages over the conventional techniques, such as the ability to generalize, parallelism and the possibility of learning, generating accurate values as the ones presented in this study.
Nevertheless, even if the presented statistical criteria of adjustment quality are good indicators for the selection of any type of model, the graphical analysis of residues is a key role to support them or not (Draper and Smith, 1981), since trend errors may occur in a given range of the response variable class, and not detected by the statistics that assess accuracy.
Hence, Figure 4 shows the graphs of network model behavior (ANN1 and ANN2) for individual wood volume estimate of eucalyptus and the real values (A), residue distribution in percentage (B) and the error frequency histogram for the two distinct settings of trained networks (C).Both networks were similar (Figure 4) having adequate predictive behavior; however, ANN1 (DAP and Ht) was superior, being more flexible in gathering data (A).Residual graph showed adequate distribution of errors with no trends in all the different classes of diameter with maximum errors within ±30%, but ANN1 presents a slightly more compact and homogeneous residue distribution (B).According to Campos and Leite (2013), the assessment of residues through histograms (C) is an interesting type of analysis, since when there are large numbers of observations, only the scatter plots may lead to a risk of misinterpreting because there are many overlapping points within the graph.In this regard, both trained networks (ANN1 and ANN2) had adequate frequency errors, with the vast majority in classes ranging between -10 and 10% error; however, it was evident that ANN1 had no error greater than ± 20%.
Based on statistics indicating accuracy shown in Table 2, the compliance between the residual distribution and histograms errors can be highlighted (Figure 4).Therefore, ANNs were able to predict accurately the individual wood volume of E. urophylla trees, using diameter and height attributes.

Neural network validation
The trained networks (ANN1 and ANN2) validation statistically proves the viability of using artificial intelligence (IA) tools to predict individual wood volume of E. urophylla (Table 3).For analysis of these statistics, 474 trees (72%) from experimental data which were not used for network trainings were used.It is therefore in accordance with Zucchini (2000), who reported that the validation sample must be independent.Additionally, meeting the modeling principles recommended by Gujarati and Poter (2011), set that at least 20% of samples that integrates the database, should be left to validate the models.
As the Shapiro-Wilk (1965) normality test presented a p-value of 1.97e -12 , which denotes non-normality of data distribution, both ANNs were submitted to the Kruskal-Wallis' test (1952) for validation, besides comparisons with each other and, in sequence subjected to the aggregated difference test in percentage (Table 3).Both ANN1 as ANN2 obtained individual wood volume values near each other, as well as close to the real values.Such proximity is detected by the Kruskal-Wallis analysis.This fact indicates that both ANNs are valid and reliable in estimating this variable (volume) using as predictors, attributes such as diameter and height.It is mentioned also that ANN1 (DBH and Ht) and ANN2 (DBH) did not differ (p> 0.05), so both can be used.
The aggregate difference (AD), which is the difference between the sums of the observed values with estimated   values, serves as an indicator criterion of over-or underestimates.In both trained networks, the AD had close and negative values (ANN1, -0.75% and ANN2, -1.75%), indicating a slight overestimation of individual volume of trees (Table 3).These low values together with the results of Kruskal-Wallis' analysis demonstrate the adherence of these neural networks to modelling the forest volume.Neural networks are nowadays a promising tool used in multidisciplinary researches (Silva et al., 2010).In Brazil, ANN use has been gaining attention for forest stand estimates, being considered an efficient and promising technique by several researchers (Leite et al., 2011;Castro et al., 2013;Binoti et al., 2015;Miguel et al., 2015).Adherence capacity, compact and homogeneous residues as well as distribution of errors within class ranges near zero are desirable in independent validation of modeling technique.These characteristics demonstrate the ability of models to estimate the variables of interest with accuracy.As in training, both ANNs had effectiveness in estimating the volume of individual trees, showing adhesion, compaction and homogeneity of residues, as the error histogram showed the highest rate close to zero (Figure 5).It is also noted that both networks (ANN1 and ANN2) had overestimation and underestimation errors of about 30%, although the biases arising from ANN2 always had values closer to the extreme.
Therefore, these results corroborate the claims of Egrioglu et al. (2014) that mentioned that ANNs have advantages over conventional techniques due to its generalizability, parallelism and the possibility of learning.Thus, the same nets can extract standards from a particular database and reapply it to other accurately, and then its use recommended.
Traditionally, in the routine of Brazilian forest inventories to measure tree volumes, it is very common to use volumetric equations of double entry, whose volume is estimated by relating the diameter at breast height (DBH) and height (Ht) of the trees.However, measuring the height of trees in forest stands is a costly activity as compared to the diameter measurement.Furthermore, the difficulty to obtain it makes impracticable the process, as some problems may occur as a lack of top of the tree visibility in dense stands, as well as the occurrence of winds, especially in eucalyptus plantations (Binoti et al., 2013).
As the Kruskal-Wallis' test showed adherence for both ANNs in predictions of individual tree volume, the performance of ANN2, which is formed by a single neuron in the input layer (diameter) is noteworthy.Therefore, this result is of great value, as it affects positively by reducing the time and cost of forest inventories, allowing accurate indirect estimates of the volume of trees in the forest, using only the diameter as a predictor variable.
However, it is emphasized that the results obtained in this study are specific to the species E. urophylla, at the same age.Thus, further studies must be carried using other species at different ages, as consequence, different neural network settings and architectures should be trained.As a suggestion, the variables "species" and "age" could be used as categorical variables in the input layer, which may result in a single ANN that will be able to accurately predict the individual volume range of different species and ages of the genus Eucalyptus.

Conclusions
Artificial neural networks of the type multilayer perceptron using the diameter and the total tree height as predictor variables are accurate in the estimate of individual wood volume of E. urophylla trees and are not statistically different from the true wood volume derived from rigorous tree scaling process.Settings of multilayer perceptron networks with a single neuron in the input layer, the diameter, are as precise and accurate as networks using two neurons in the same layer, the diameter and height.

Figure 1 .
Figure 1.Study area location and spatial distribution of sampled plots.Source: Google Earth image; Natural Earth (naturalearthdata.com);IBGE (Brazilian Institute of Geography and Statistics).

Figure 2 .
Figure 2. Normal distribution probability (A) and diameter distribution in a stand of E. urophylla S. T. Blake sampled in the state of Goiás, Brazil (B).

Table 1 .
Frequency distribution of the optimal number (α = 0.05; 5%) of measured tree for training of the ANNs, by diameter and height classes for Eucalyptus urophylla in the state of Goiás, Brazil.at 1.3 m above ground, Ht = total height.

Figure 3 .
Figure 3. Architecture of artificial neural networks (ANN) for prediction of individual total wood volume of forest stands of E. urophylla, in function of the DBH and Ht (ANN1) and DBH (ANN2), in the state of Goiás, Brazil.

Figure 4 .
Figure 4. Behavior in the training of ANN1 (DBH and Ht) and ANN2 (DBH) in predicting individual wood volume (A), residual distribution (B) and the error frequency histogram (C) for stands of E. urophylla in the city of Rio Verde, state of Goiás, Brazil.

Figure 5 .
Figure 5. Behavior in validations of ANN1 (DAP and Ht) and ANN2 (DAP) in predicting individual wood volume (A), residual distribution (B) and error frequency histogram (C) for forest stands of E. urophylla in the city of Rio Verde, state of Goiás, Brazil.

Table 2 .
Characteristics and performance statistics of the artificial neural networks (ANN) selected to estimate the volume of E. urophylla trees in the city of Rio Verde, state of Goiás, Brazil.
IT = Levels of training (network acquisition), IS = levels of stop selection (training stop), IA = levels of assessment (trained network quality), ̂ = correlation between observed and predicted volume, RMSE% = root-mean-square error in percentage.

Table 3 .
Mean, minimum and maximum values, as well as real and estimated values of individual wood volume, estimated by both network categories, and validation statistics of E. urophylla in Rio Verde, state of Goiás, Brazil.