Accurately estimating the volume of trees has always been a very important interest for forest managers (Tewari and Kumar 2001). For instance, stem volume estimation for standing trees is essential in the assessment of forest biomass. According to several authors (Fang and Wang 2001, Lehtonen et al. 2004, Vallet et al. 2006, Tobin and Nieuwenhuis 2007), volume estimates can be used to determine the biomass of the whole tree by using stem density and biomass expansion factor. Furthermore, stem volume estimates are required for sustainable forest management (Hofstad 2005, Haywood 2009). They are also essential for sustainable forest resources planning (Zianis et al. 2005), assessment (Guendehou et al. 2012) and environmental studies (Hofstad 2005). Stem volume can be estimated with field measurements of diameter at breast height (DBH) and total height and by allometric equation development (Teshome 2005, Akindele and LeMay 2006, Gonzalez-Benecke et al. 2014). According to Avery and Burkhart (2002), volume equations are used to estimate the average content of standing trees of various sizes and species.

Jeju Island has a total land area of 184,840 ha and approximately 88,874 ha (48%) of this island has a forest cover (Korea Forest Service 2012). Jeju Island is internationally recognized for its biodiversity. It was designated as a biosphere reserve by the United Nations Educational, Scientific and Cultural Organization on December 2002 (UNESCO 2014). One of the most important forests in this island is the Gotjawal forest, which is conserved and protected due to its unique ecosystem. One such ecosystem is the Dongbaekdongsan wetland, which is designated as a National Wetland Protected Area by the Ministry of Environment of the Republic of Korea and classified as a Ramsar site (Ramsar Convention 2014). Gotjawal forest has a total land area of 11,000 ha, accounting for 6% of the total land area of Jeju Island (Kang et al. 2013) and is situated on the middle slopes of Mount Halla, the highest mountain in the Republic of Korea. Furthermore, Kang et al. (2013) stated that this forest has very significant roles in establishing biological and cultural diversity while maintaining its ecosystems. One of the most dominant tree species in the forest is Quercus glauca Thunb. (Han et al. 2007, Oh 2012, Kang et al. 2013). However, the volume equation for Quercus glauca in Jeju Island, which is essential in accurately estimating the biomass and carbon stocks of Gotjawal forest, has not yet been developed. Thus, the objectives of this study were to develop and to validate volume equations for Quercus glauca in Jeju Island, Republic of Korea.

Jeju Island is located in the southernmost part of the Republic of Korea, between 126°08′43″ to 126°58′20″E and 33°11′27″ to 33°33′50″N (Lee et al. 2009) as shown in Fig. 1. Broadleaved forests comprise the majority of the forestland of Jeju Island with 30,538 ha. The study site is located in the Gotjawal forest, which is classified as a subtropical forest (Korea Forest Research Institute 2014). The elevation of the sampling areas ranges from 86-235 m above sea level. The mean annual temperature (MAT) of the island is 15.80℃. In addition, the minimum MAT is 12.90℃ and the maximum MAT is 18.90℃(1981-2010). The mean annual precipitation is 1,497.60 mm (Korea Meteorological Administration 2014).

The DBH (cm) and total height (m) of the 167 sample trees of Quercus glauca were measured. The DBH was measured using a standard diameter tape and the total height was measured using a Haglöf Vertex III Transponder and Transponder (Haglöf, Sweden). Because cutting or harvesting of Quercus glauca trees is strictly prohibitedin Gotjawal forest due to conservation restrictions and destructive sampling of trees is very expensive, labor intensive and time consuming, the diameter outside the bark (DOB, cm) along the stem and the corresponding heights from the ground (m) were measured using the CRITERION400 laser dendrometer (Laser Technology, United States). The DOBs along the stem of each tree were determined starting from 20 cm above the ground and incrementing by 1-2 m. Using the Smalian formula, the volume of the different stem sections were determined. For the last upper stem section, the conical formula was used. The volumes of the different log sections were summed up to calculate the stem volume of each tree. To examine the relationship of DBH - volume of Quercus glauca, the stem volume was plotted versus the DBH (Fig. 2). The dataset was randomly split into two subsets. For the initial model development, 80% of the dataset was used while the remaining 20% was used for model validation (Table 1).

Five common volume equation forms used in various forestry literatures (Clutter et al. 1983, Avery and Burkhart 1994, van Laar and Akça 1997) were selected as candidate models to estimate the stem volume outside the bark of Quercus glauca in Jeju Island (Table 2). These models have been used in several studies for different species in various countries (Tewari and Kumar 2001, Kozak and Kozak 2003, Tewari and Kumar 2003, Dela Cruz and Bruzon 2004, Segura and Kanninen 2005). The parameters of these models were estimated using the Statistical Analysis System Non-Linear (SAS NLIN) procedure (SAS Institute Inc. 2004). In order to evaluate the performance of these models, the standard error of estimate (SEE), mean bias absolute mean deviation (AMD), coefficient of determination (R²), and root mean square error (RMSE), were determined (Table 3). To identify the best models, rank analysis was used. The model with the lowest values for SEE, AMD and RMSE had the best rank. For R², the model with a value closest to 1 had the best rank, whereas for a value close to zero was considered best.

Using 80% of the dataset, the initial volume model development was completed with five candidate volume equations. The results of the evaluation showed that the SEE ranged from 0.02099 to 0.03240, with model 5 (Schumacher and Hall 1933) having the best value and model 1 the poorest. The AMD ranged from 0.01354 m³ (model 5) to 0.01995 m³ (model 1), R² ranged from 0.92234 (model 2) to 0.98446 (model 5) and RMSE ranged from 0.02098 (model 5) to 0.03240 (models 1 and 2). Model 5 had the highest over prediction with value of -0.00112 m³, while model 4 had the highest under prediction with value of 0.00534 m³. Based on the evaluation and rank analysis, model 5 was considered the best model for predicting the stem volume of Quercus glauca in Jeju Island.

To better determine which model provided the most accurate estimation of stem volume, model validation was conducted. The AMDs of the five candidate models in the different DBH classes (5 cm interval) were determined as shown in Fig. 3. The validation results showed that model 5 had the best AMD in most of the DBH classes, while models 1 and 2 (with only DBH as predictor) had the poorest performance among the five candidate models.

The two datasets were combined for final modeling and evaluation. The parameter estimates of the five candidate models were estimated and are shown in Table 4. Using the five fit statistics, the performance of these models were evaluated as shown in Table 5. Model 5 still provided the best SEE with 0.02745, AMD with 0.01538 m³, R² with 0.97603 and RMSE with 0.02746. For the value, model 4 had the highest under prediction (0.00562 m³) and model 5 had the highest over prediction (-0.00129 m³). Models with one predicting variable (DBH) had the poorest performance in most of the fit-statistics. Model 1 had the highest SEE with 0.03764, AMD with 0.02275 m³ and RMSE with 0.03768, while model 2 had the lowest R² with 0.90484. Using rank analysis, the best model was found to be model 5, while model 1 was the poorest.

Kozak and Kozak (2003) recommended using lack-of-fit statistics to determine further which of the candidate models is the most suitable for predicting stem volume. In this evaluation, one or more evaluation statistics must be determined for the various subgroups of the independent variable utilizing the entire dataset. In this study, SEE,   and AMD in the different DBH classes (5 cm interval) for each model were determined. Results showed that model 5 consistently performed the best among the candidate models, with the best SEE, and AMD in most of the DBH classes, while model 1 ranked last (Table 6). This evaluation method is important because a single overall value from the fit statistics may not indicate which model is best (Kozak and Smith 1993). For instance, in the negative and positive biases of a model may have a cancelling effect that may produce a better (close to zero) (Berhe and Arnoldsson 2008).

Overall, models with DBH and height as predictors had better performance than the two models with only DBH as a predictor. This result is comparable to other studies (Tewari and Kumar 2001, 2003, Guendehou et al. 2012) which showed that models with two predictors (DBH and height) had better predictive capabilities than those with only one predictor (DBH) based on the fit statistics.

Although models 1 and 2 had lower performance in stem volume prediction, these two models are still effective, especially when the total height of the tree is unavailable. For instance, the Korea Forest Service conducted the 5th National Forest Inventory (NFI) from 2006-2010 in the Republic of Korea and only 10 representative trees in a plot were measured for total height due to the difficulty and high cost of height measurement, whereas the DBH of all of the trees were measured. Furthermore, Segura and Kanninen (2005) recommended developing volume models using only DBH as a predictor because it can be measured easily and accurately in the field, unlike the total height.

Stem volume models with two predictors, DBH and total height, showed better predictive capability compared to the two models with only one independent variable (DBH) when estimating the volume outside the bark of Querucs glauca in Jeju Island, Korea. Model 5, originally proposed by Schumacher and Hall (1933), is considered as the best based on model evaluation and validation. On the other hand, model 2 is recommended in instances where total height is unavailable. Model 5 also provided a more accurate stem volume prediction than the general volume function in Forest Resource Evaluation and Prediction Program for broadleaf species. It is hoped that the volume models of the present study can help forest managers in sustainably managing the Quercus glauca forests in Jeju Island as these models can provide more reliable estimates of the growing stock stem volume.