The Effect of Crown Dimensions on Stem Proﬁle for Dahurian Larch, Korean Spruce, and Manchurian Fir in Northeast China

: Crown architecture has long been evaluated for its impact on taper modeling. However, most of the research has focused on a limited number of crown dimensions. This study examined the effect of adding several crown dimensions in improving the diameter and volume estimates of Dahurian larch, Korean spruce, and Manchurian ﬁr in northeast China. The crown dimensions included crown length, crown ratio, crown width, height to live crown base, diameter at the crown base, and crown shape. A well-known taper model of Clark et al. (1991) was ﬁtted to the data of 276 trees from natural stands. To adjust the inherent autocorrelation in the data, we added a third-order continuous-time error structure in the model ﬁt. Model ﬁtting was carried out with the NLMIXED procedure (Non-linear Mixed Procedure), followed by the MODEL procedure of SAS using the generalized nonlinear least-squares method. Fit statistics and graphical assessments were used to evaluate the original and modiﬁed models. Above 98% of the total variance of d was explained by the models for all species. The addition of crown variables showed slight improvements for root mean square error (RMSE) values in the analyzed species. The RMSE plots indicated that the models with crown variables slightly improved the diameter and volume predictions for the species but only for the upper stem (>50%–90%). The study demonstrated that crown dimensions inﬂuence the stem taper, but the original model of Clark et al. (1991) reasonably realized that effect. The models with CL and behaved inconsistently for and upper stem sections of Korean and Dahurian larch, the models performance for basal and stem (relative heights, they for upper stem sections. In the models with crown variables performed better than the original model for the relative heights of The model with CL showed the best results, which was followed by the model with CR for Dahurian larch.


Introduction
The forests of northeast China represent about 30% of the total forest area in China and serve as a national base of wood products. Additionally, almost half of the national ecosystem carbon is stored in these forests [1]. NE China hosts different forest types, ranging from temperate broadleaf forest to boreal taiga forest. Coniferous forest mainly includes Dahurian larch (Larix gmelinii Rupr.), Manchurian fir (Abies nephrolepis), Korean spruce (Picea koraiensis Nakai), and Korean pine (Pinus koraiensis Siebold & Zucc.). The region falls under the boreal continental climate and is characterized by the southern border of the discontinuous permafrost zone [2,3].
Dahurian larch, Korean spruce, and Manchurian fir are valuable conifer species with a corresponding volume of 955.2 million m 3 , 1001.6 million m 3 , and 1135.6 million m 3 in this region [4]. These species provide plywood, veneer, and high-quality timber for construction, railway sleepers, and shipbuilding. Other uses include soundboards for musical instruments, boxes, and raw materials for the pulp industry. Besides wood products, forest management objectives in China include biodiversity conservation, soil protection, and carbon sequestration [5]. Taper models have been used to estimate tree volume and biomass simultaneously, which allows for extending the timber inventories to ecological studies [6]. Accurate growth and yield modeling is a critical step in this direction to know the potential productivity of natural forests [7].

Data Description
A sample of 276 trees was collected from uneven-aged natural stands, representing the existing range of diameter and height classes in the study area. The sample comprised 109 trees of Dahurian larch, 103 trees of Korean spruce, and 64 trees of Manchurian fir. Diameter at breast height over bark (D, 1.3 m) and crown width (CW) were recorded to the nearest 0.1 cm for all trees. Crown width was calculated by taking the average of two projections of perpendicular axes from the crown base to the ground. Trees were felled to measure total tree height (H), height to the live crown base (HCB), and stem diameter over bark at the crown base (CD). The live crown base was considered to be at the lowest branch whorl. Diameters over bark (d) were measured at the heights (h) of 0.3, 0.6, 1, 1.3, and 2 m. After the height of 2 m, d was measured at a fixed interval of 1 m. The measurement range was 0.3 to 1 m along the stem except for the top section, which was considered as a cone. Two perpendicular diameters (over bark) were measured, and their average was used. Smalian's formula [37,38] were used to calculate log volumes that were added to the volume of the cone to get over bark total volume. Finally, crown length (CL), crown ratio (CR), and crown shape (CS) were derived using the crown measurements. The ratio between crown length and total tree height determined the crown ratio. The ratio between crown width and crown length provided the crown shape. The summary statistics of the datasets are shown in Table 1.
The model of Clark comprises Schlaegel's form-class model and Max and Burkhart's segmented model.

Model Derivation
Initially, the fitting of Clark's model was carried out using tree-level mixed-effect modeling (Non-linear Mixed Procedure (NLMIXED procedure) in SAS). The general mixedeffects model of Clark is where µ 1 , µ 2 , µ 3 , µ 4 , µ 5 , µ 6 are random parameters. All other variables are the same as described earlier.
After identifying the best combinations of random-and fixed-effect parameters, we substituted the random parameters with crown variables [28]. For the selection of best combination, the criteria were (1) Akaike's Information Criterion (AIC), (2) Schwarz's Bayesian Information Criterion (BIC), and (3) twice the negative log-likelihood (−2Ln (L)).

Parameter Estimation/Model Fitting
In the final step, seven models were fitted: The Clark model with and without crown variables (CL, CR, CW, HCB, CD, and CS). Mixed-effects modeling was not used at this stage because it requires calibration for improving the diameter estimates. Mixed-effects modeling was only carried out in the initial step to identify the random parameters [28]. The model parameters were estimated with the MODEL procedure of SAS using the generalized nonlinear least-squares method [38]. The spatial correlation was expected within the observations due to the hierarchical data of the study. We instituted a third-order continuous autoregressive error structure (CAR (3)) to adjust the innate autocorrelation. This specified error structure allows the practical use of a model for irregularly spaced and unbalanced data [42]. Programming for CAR (3) structure was worked out in the MODEL procedure of SAS [38].

Model Comparisons
The accuracy of the models was evaluated by numerical and graphical assessments of diameter and volume predictions. The measured diameters were used to calculate sectional volumes, which were added to obtain the observed total volume. Similarly, the predicted diameters were utilized to calculate the predicted total volume. Stump height was 0.3 m and the length of sections varied from 0.3 to 1 m along the stem in this analysis. The observed volume of each section was calculated by Smalian's formula, while the predicted volume was calculated using the volume equation of Clark et al. (1991). Two standard goodness-of-fit statistics were tested: root mean square error (RMSE) and fit index (FI). The notations for these statistics are as under; where y i ,ŷ i , and y stand for measured, predicted, and average values of the response variable; n symbolizes the total number of observations, and p is the number of parameters, respectively. The accuracy of the models was also assessed for different stem sections. The stem was divided into ten sections (relative heights), and RMSE was calculated for diameter and volume in each section. The RMSEs were plotted against relative heights to compare the models' performance in predicting diameter and volume. Taper simulations were generated to analyze the influence of different crown variables on taper formation of Dahurian larch, Korean spruce, and Manchurian fir.

Results
We fitted Model 2 of Clark with possible combinations of fixed-and random-effects parameters. The fitting of the model with more than two random-effects parameters failed to achieve convergence. Therefore, we tested this model with random-effects added to one or two fixed-effects parameters only. Table 2 shows the evaluation statistics for the model with the best one and two random-effects parameters, as well as fixed-effects models. The mixed models with two random parameters, u2 and u6, showed the lowest values of AIC, BIC, and −2Ln (L).
The Clark model with the random parameters, u 2 and u 6 , substituted by crown variables, resulted in the following model 6.
where λ i are the parameters and crwn represents the crown variables, which denoted either CL, CR, CW, HCB, CD, or CS. All other variables are the same as described earlier. Model 1 and model 5 with different crown variables were fitted using the generalized nonlinear least-squares method. The initial fitting of the models without the error structure resulted in strong autocorrelation. As an example, the autocorrelation observed in the d predictions with the Clark model is shown in Figure 1. This correlation trend disappeared when a third-order continuous autoregressive error structure CAR (3) was added in the model fit. where λi are the parameters and crwn represents the crown variables, which denoted either CL, CR, CW, HCB, CD, or CS. All other variables are the same as described earlier. Model 1 and model 5 with different crown variables were fitted using the generalized nonlinear least-squares method. The initial fitting of the models without the error structure resulted in strong autocorrelation. As an example, the autocorrelation observed in the d predictions with the Clark model is shown in Figure 1. This correlation trend disappeared when a third-order continuous autoregressive error structure CAR (3) was added in the model fit.  Table 3 shows the parameter estimates, and Table 4 highlights the fit-statistics of the models with and without crown variables for the species. Above 98% of the total variance of d was explained by the models for all species. The addition of crown variables showed slight improvements for FI values in the analyzed species. Inclusion of crown variables reduced the RMSE from 0.004 cm to 0.031 cm for Dahurian larch, −0.0001 cm to 0.001 cm for Korean spruce, and 0.0 cm to 0.006 cm for Manchurian fir, respectively. This reduction was more evident in the models with CL (2.63% and 0.39%) for Dahurian Larch and Korean spruce and CR (2.29%, 0.39%, and 1.13%) for Dahurian larch, Korean spruce, and Manchurian fir, respectively. The improvement in RMSE was marginal (0.40-1.13%) for Korean spruce and Manchurian fir.  Table 3 shows the parameter estimates, and Table 4 highlights the fit-statistics of the models with and without crown variables for the species. Above 98% of the total variance of d was explained by the models for all species. The addition of crown variables showed slight improvements for FI values in the analyzed species. Inclusion of crown variables reduced the RMSE from 0.004 cm to 0.031 cm for Dahurian larch, −0.0001 cm to 0.001 cm for Korean spruce, and 0.0 cm to 0.006 cm for Manchurian fir, respectively. This reduction was more evident in the models with CL (2.63% and 0.39%) for Dahurian Larch and Korean spruce and CR (2.29%, 0.39%, and 1.13%) for Dahurian larch, Korean spruce, and Manchurian fir, respectively. The improvement in RMSE was marginal (0.40-1.13%) for Korean spruce and Manchurian fir.  The plots of RMSE against relative heights providing the diameter and volume predictions of the models are displayed in Figures 2-4. All models indicated similar diameter predictions along the entire stem for Korean spruce and Manchurian fir. However, for volume predictions, the models with HCB and CR showed slightly better results for 50-90% relative heights of Korean spruce. The models with HCB and CR accompanied by the model with CL marginally improved the predictions for 40%-60% relative heights of Manchurian fir. The models with CL and CD behaved inconsistently for middle and upper stem sections of Korean spruce and Manchurian fir, respectively. For Dahurian larch, although the models showed similar performance for the basal log and middle stem (relative heights, 0%-50%), they behaved differently for upper stem sections. In general, the models with crown variables performed better than the original model for the relative heights of >50%-90%. The model with CL showed the best results, which was followed by the model with CR for Dahurian larch. predictions along the entire stem for Korean spruce and Manchurian fir. However, fo volume predictions, the models with HCB and CR showed slightly better results for 50 90% relative heights of Korean spruce. The models with HCB and CR accompanied by th model with CL marginally improved the predictions for 40%-60% relative heights of Man churian fir. The models with CL and CD behaved inconsistently for middle and uppe stem sections of Korean spruce and Manchurian fir, respectively. For Dahurian larch, al hough the models showed similar performance for the basal log and middle stem (relativ heights, 0%-50%), they behaved differently for upper stem sections. In general, the mod els with crown variables performed better than the original model for the relative heigh of >50%-90%. The model with CL showed the best results, which was followed by th model with CR for Dahurian larch.          Taper simulations were also generated for the trees of the same size with different values of crown variables (Figures 5-7). Crown variables, irrespective of lower or higher values, did not affect the taper of Korean spruce, whereas the higher values of CD and CS slightly reduced the taper of Manchurian fir. Dahurian larch was more sensitive to crown variables, which influenced the taper differently. Higher values of CL, CR, CD, and CW produced greater taper, although the latter variable showed a modest influence. Inversely, higher HCB and CS showed less taper.
values of crown variables (Figures 5-7). Crown variables, irrespective of lower or higher values, did not affect the taper of Korean spruce, whereas the higher values of CD and CS slightly reduced the taper of Manchurian fir. Dahurian larch was more sensitive to crown variables, which influenced the taper differently. Higher values of CL, CR, CD, and CW produced greater taper, although the latter variable showed a modest influence. Inversely, higher HCB and CS showed less taper.

Discussion
Most of the past taper analyses accounted for a limited number of crown dimensions for different species in the world. The commonly used crown variables are CL and CR [18,22,26,27]. Reference [8] incorporated HCB along with CR and CL in their study. This analysis evaluated the role of CL, CR, CW, HCB, CD, and CS in improving the taper models for Dahurian larch, Korean spruce, and Manchurian fir in NE China.

Discussion
Most of the past taper analyses accounted for a limited number of crown dimensions for different species in the world. The commonly used crown variables are CL and CR [18,22,26,27]. Reference [8] incorporated HCB along with CR and CL in their study. This analysis evaluated the role of CL, CR, CW, HCB, CD, and CS in improving the taper models for Dahurian larch, Korean spruce, and Manchurian fir in NE China.
The initial fitting of the Clark model was carried out as a tree-specific mixed model with possible combinations of fixed-and random-effects parameters. The fitting of the model with more than two random-effects parameters failed to achieve convergence. Therefore, we tested this model with random-effects added to one or two fixed-effects parameters only. The mixed models with two random parameters, µ 2 and µ 6 showed the lowest values of AIC, BIC, and −2Ln (L).
Although the addition of random parameters µ 2 and µ 6 improved the fitting, the models expanded with both µ 2 and µ 6 provided the best statistics. The addition of random parameters µ 2 and µ 6 affects the lower section and middle/upper section, respectively. The original equation is quadratic for the lower portion and linear for the upper portion [34]. The adapted model with two random parameters allows for the variation in both segments between trees. The expansion of the models with random effects µ 2 and µ 6 is tenable, considering the effects of parameters on different stem sections. The parameters b 1 -b 3 correspond to the basal log (<1.3 m), b 4 to the lower stem (1.3-5.3 m), and b 5 -b 6 to the middle and upper stem (>5.3 m). The random-effects affected all parameters b 1 -b 4 , but the parameter b 2 showed the best statistics among different combinations. The variation was usually minimum in the lower stem (see, for example, [34], Figure 3, Table 4). The parameter b 2 explained the basal log and compensated for the slightly tapered lower stem. The parameter b 6 captured the variation in the middle stem and joined b 5 to control the upper stem. With a higher value, parameter b 6 had a major impact on the model, and parameter b 5 contributed a little, indicating that parameter b 6 explained the tree-level variation in both sections.
According to the past literature, the potential of crown variables in improving the predictions of a taper model depends on the specific species, and it might be studyspecific. The magnitude of improvement depends on natural changes in crown architecture, variations in crown sizes in available data, and the response of a particular taper model to crown variables [8].
Reference [18] found that the addition of CL and CR in the Max and Burkhart (1976) model improved diameter predictions for the middle stem (30%-70%) of loblolly pine. Reference [26] added CR in the model of Clark et al. (1991) for estimating the diameter and volume of yellow poplar. The modified model was slightly less accurate for the relative heights of 30%-50%, while it was marginally better for upper sections (>50%-60%). Both models, modified and original, showed accurate predictions for the lower stem (<30%). Reference [27] obtained a similar performance from the model of Clark with CR and CL for black pine. Reference [28] observed that the Max and Burkhart (1976) model with CR decreased the RMSE by 10% for diameter estimates of Dahurian larch. However, improvement in the values of FI was nominal. Reference [8] added CL, CR, and HCB to the model of Clark and received a mixed response for balsam fir, red spruce, and white pine. The variables of CL and CR improved the diameter predictions of red spruce and balsam fir, while HCB showed a positive impact only for balsam fir. However, the improvements were modest since the largest reduction in RMSE was 0.03 cm for balsam fir. The Clark model with CR and CL slightly better predicted the volume of all species, but the addition of HCB negatively affected the predictions for red spruce and white pine.
In agreement with previous studies, the addition of crown variables improved the prediction accuracy for the species analyzed, although noticeable improvements were only observed for Dahurian larch. Slightly different from [26,27], the models with or without crown variables were equally suitable for the lower stem (<50%) of Dahurian larch, while they showed similar performance for all stem sections of Manchurian fir and Korean spruce. Similar to [8], the models with CR and CL performed better than the models with HCB in this analysis. However, we did not find a negative effect of HCB or other crown variables across the species except a minor distortion in CL and CD for Korean spruce and Manchurian fir, respectively. Reference [8] suggested that shade tolerance could be an important factor in taper modeling with crown variables. Their suggestion might transfer to this analysis, given that the analyzed species have different levels of shade tolerance. The crown morphology of shade-intolerant species is different from the species that are tolerant of shade. The former produces a lower stem taper and live crown ratio [43]. Taper simulations in this study also reflected variation in stem taper among the species.
Although the effect of crown architecture on stem form has long been documented [16,44], its inclusion in taper modeling has varied. Some researchers found that the crown variables were either nonsignificant or marginally improved model fits (8,20,22). In contrast, some studies have observed otherwise, suggesting that it might be influenced by the available dataset, species, and model form [18,45,46]. However, [31] noted that the addition of crown variables should not only be to improve model predictions but also to ensure the logical biological behavior of the model. Therefore, a taper equation should predict trees of smaller crowns compared with the trees of similar size having bigger crowns to allow for wider diameters above diameter at breast height (DBH), thinner diameters below DBH, and a similar DBH. So, the persistent analyses of incorporating crown variables are still essential and encouraged [47]. It might include further analysis of crown variables for a wide range of species and stand conditions, besides the influence of various measures of the crown. For example, the commonly used variable is the crown ratio, whereas height to crown midpoint [48], crown length [27], or a combination of crown variables could also be effectual [18].

Conclusions
This study used several crown variables in an initial attempt to improve the model of Clark et al. (1991) for Dahurian larch, Korean spruce, and Manchurian fir in Northeast China. The addition of crown variables modestly improved the predictions for Dahurian larch and showed a minimal improvement for Manchurian fir and Korean spruce. The model of Clark et al. (1991) without crown variables demonstrated quite reasonable predictions for diameter and volume in terms of fit-statistics and graphical analysis for different stem sections. Different crown variables influenced the taper of Manchurian fir and Korean spruce least. However, for Dahurian larch, larger CL and CR showed more taper for the upper stem and near the ground, while HCB and CS indicated inverse relations for these sections. Although incorporating crown variables improved the model performance, the gain was too little to justify the additional measurements.
It is worth noting that the analyzed species are widely distributed in Northeast China. This region has significant variations in terms of geo-climatic factors, which are reflected in crown morphology. Future analysis of a sample with a larger representation could extend the scope and generalize the results of this study.
Author Contributions: A.H., investigation, software, formal analysis, writing-original draft preparation, visualization; M.K.S., methodology, investigation, writing-review and editing; L.J., Conceptualization, software, validation, resources, data curation, supervision, project administration, funding acquisition. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.