Stem taper functions for white birch ( Betula platyphylla ) and costata birch ( Betula costata ) in the Xiaoxing’an Mountains, northeast China

White birch ( Betula platyphylla Sukaczev) and costata birch ( Betula costata Trautv.) are valuable hardwood tree species growing in northeast China. Several taper studies have analysed birch species in the countries harbouring the boreal forests. However, this study presents an initial attempt to develop stem taper models using the ﬁxed-and mixed-effects modelling for white birch and costata birch in Xiaoxing’an Mountains, northeast China. Ten commonly used taper models were evaluated by using 228 destructively sampled trees of both tree species comprising of 4582 diameter and height measurements. The performance of these models was tested in predicting diameter at any height, total volume and merchantable volume (10 and 20 cm top diameters). We incorporated a second-order continuous-time error structure to adjust the inherent autocorrelation in the data. The segmented model of Clark best predicted the diameter and total or merchantable volume when the upper stem diameter at 5.3 m was available. When diameter measurements at 5.3 m were not available, the models of Kozak and Max and Burkhart were superior to other models for white birch and costata birch, respectively. After model comparison, the best model of Clark was reﬁtted as the NLME model.


Introduction
Forest area in China accounts for 220 million ha with a standing volume of 17.56 billion m 3 , as per the latest National Forest Inventory (NFI-9). Forest cover has expanded from 12.69 to 22.96 per cent during the last 40 years. Sustainable forest management, however, remains a rigorous undertaking to meet national and international commitments. State Forestry Administration aims to maintain the forest cover at over 26 per cent by the year 2050 to bridge the supply and demand gap and achieve a sound cycle of ecological conditions (Xu et al., 2019). A critical step in this direction is accurate growth and yield modelling to know the exact potential productivity of natural forests (Hou et al., 2019).
Birch species naturally grow in East Asia, Scandinavia, Baltic States, Russia and North America. White birch (Betula platyphylla Sukaczev) and Korean/costata birch (Betula costata Trautv.) are major valuable tree species of China covering an area of 10.38 million ha, with a total volume of 923 million m 3 (Xu et al., 2019). The greatest area under these species lies in northeast China. White birch is the principal source for plywood, paper pulp and furniture industry. Recent studies indicate that it has a high medicinal value (Xu et al., 2016;Wang et al., 2019). Costata birch provides the raw material for pulp and fibre-based products. They play an important role in maintaining the ecological balance of the natural broad-leaved secondary forest in Northeast China (Wang et al., 2018a;Zhao et al., 2019).
Stem taper models are an essential component of growth and yield modelling. Taper models can predict the stem diameter at any height (d), along with merchantable and total volumes (Li and Weiskittel, 2010). These models supersede the volume tables as they can estimate d, merchantable height to any diameter above ground, the volume of a log at any length and at any height from the ground in addition to the merchantable and total stem volume (Kozak, 2004). Additionally, taper models are useful in timber quality studies and modelling of carbon allocation in different stem sections. They are also instrumental in assessing the impact of silvicultural treatments on stem taper (Fonweban et al., 2011).
Since the last century, many taper models have been developed. At present, an in-depth discussion is available about their evolution and types (Sakici et al., 2008;Crecente-Campo et al., 2009;Burkhart and Tomé, 2012). Of the many model forms, segmented or variable form taper models are often recommended based on taper studies (Özcelik and Brooks, 2012;Gómez-García et al., 2013;Sakici and Ozdemir, 2018). Rojo et al. (2005) and Tang et al. (2017) suggested the variable form taper models Downloaded from https://academic.oup.com/forestry/advance-article/doi/10.1093/forestry/cpab014/6189774 by guest on 29 March 2021 Forestry for diameter estimates of maritime pine (Pinus pinaster Ait.) in Spain and Himalayan birch (Betula alnoides Buch.-Ham. ex D.Don.) in South China, respectively. Alternatively, segmented taper models performed better than variable form models for diameter and volume estimates of Scots pine (Pinus sylvestris L.) in northwestern Spain (Diéguez-Aranda et al., 2006) and Lebanon cedar (Cedrus libani A. Rich.) and Cilicica fir (Abies cilicica Carr.) in Bucak region, Turkey (Özcelik and Dirican, 2017). Simultaneously, the performance of segmented and variable form models was very similar for Kazdagi fir (Abies nordmanniana subsp. equitrojani (Asc. & Sint. ex Boiss.) Coode & Cullen) and Oriental beech (Fagus orientalis Lipsky) in Turkey and white birch (B. platyphylla Sukaczev) in northeast China (Sakici and Ozdemir, 2018;Shahzad et al., 2019Shahzad et al., , 2020. Therefore, it is useful to perform a thorough analysis of these taper models so that their application can be extended to other species. The research on taper modelling has mostly focused on conifers species with an excurrent stem form. Generally, hardwood tree species exhibit a decurrent crown. This spreading crown constrains the merchantability of the main stem, in particular, for the saw timber. Hardwood trees of similar total heights have different merchantability limits (Fowler and Rennie, 1988). However, very few authors (e.g. Martin, 1981;Westfall and Scott, 2010) have evaluated taper models in estimating volume for different merchantable products (sawlog, pulpwood, etc.). This practical consideration was accounted for in the present analysis.
Over the years, mixed-effects models have gained attention in taper modelling (Trincado and Burkhart, 2006;Yang et al., 2009;Cao and Wang, 2011;Özcelik et al., 2011;Zhao and Kane, 2017). Compared with ordinary least squares, mixed models contain both fixed effects and random effects parameters to explain between-tree and within-tree variations in taper studies. Furthermore, this technique enables the calibration of a model for a particular tree or site, if additional data are available for that specific tree or site (Garber and Maguire, 2003;de-Miguel et al., 2013;Cao and Wang, 2014).
Most of the existing studies used mixed-effects modelling to fit a single model, either variable form (Lejeune et al., 2009;Arias-Rodil et al., 2015;Bronisz and Zasada, 2019) or segmented type (Leites and Robinson, 2004;Trincado and Burkhart, 2006;Cao and Wang, 2011). The studies by de-Miguel et al. (2012) and Gómez-García et al. (2013) fitted different fixed-effects models and selected variable form models for developing a mixed model. However, the later study tested the fixed-effects models for diameter predictions only. Therefore, it is beneficial to further evaluate different taper models with both fixed-and mixedeffects approaches and extend their scope to other species.
There are many references to taper studies of birch species in the world (e.g. Kozak et al., 1969;Martin, 1981;Laasasenaho, 1982;Gál and Bella, 1994;Westfall and Scott, 2010;Gómez-García et al., 2013;Ung et al., 2013). These studies accounted for paper birch (Betula papyrifera Marsh.), yellow birch (Betula alleghaniensis Britton), sweet birch (Betula lenta L.), grey birch (Betula populifolia Marsh.), river birch (Betula nigra L.) and downy birch (Betula pubescens Ehrh.) in Canada, the US and Europe. Research on white birch and costata birch is quite active in China, but it mainly focusses on biomass and genetics (Cai et al., 2013;Wang et al., 2018b;Wang et al., 2019). Recently, Shahzad et al. (2019, 2020 tested fixed-effects models for white birch in Daxingan Mountains, Northeast China. To our knowledge, no taper study exists for white birch and costata birch that have considered both fixed-and mixed-effects modelling. The specific objectives of this study were to: (1) evaluate well-known segmented and variable form taper models using fixed-effects and select the best model for diameter and volume prediction of white birch and costata birch and (2) develop treespecific mixed-effects models based on the best taper model.

Data description
A sample of 120 trees of white birch and 108 trees of costata birch was collected from uneven-aged natural stands in the study area. Data from a total of 228 trees with 4582 diameter and height measurements were utilized. The sample covered the existing range of diameter and height classes. The range of diameter and height was 4.8-37.9 and 6.9-22.2 m for white birch and 19.0-49.2 and 13.8-23.4 m for costata birch. Diameter at breast height over bark (D, 1.3 m) was measured to the nearest 0.1 cm for all trees. Trees were felled to measure total tree heights (H) and their diameters over bark (d) 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. Measurement range fluctuated from 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 (Burkhart et al., 2019) was used to calculate log volumes that were added to the volume of the cone to get over bark total volume. Smalian's formula is V = A 1 +A 2 /L, where V is the volume of the log in m 3 , A 1 is the area of the small end of the log in m 2 , A 2 is the area of the large end of the log in m 2 and L is the length of the log in m. The formula for the volume of the cone is V T = π td T 2 /12, where V T is the volume of the terminal section, d T is the diameter at the base of the last section and t is the distance from the base of the last section to the stem tip (West, 2015). The summary statistics of the datasets are shown in Table 1. The plots of relative diameter (d/D) against relative height (h/H) for both species are shown in Figure 1.

Figure 1
The plots of relative diameter against relative height for white birch (a) and costata birch (b).

Taper models
Ten taper models were selected from the literature (Sakici et al., 2008;Li and Weiskittel, 2010;Özcelik and Crecente-Campo, 2016;Doyog et al., 2017;Tang et al., 2017). The selected models covered four segmented taper models (Max and Burkhart, 1976;Clark et al., 1991;Fang et al., 2000) and six variable form taper models (Muhairwe, 1999;Bi, 2000;Lee et al., 2003;Kozak, 2004;Sharma and Zhang, 2004;Sharma and Parton, 2009). In the following text, these models will be written as Max and Burkhart (1976), Clark et al. (1991) Model 1 and Model 2, Fang et al. (2000), Lee et al. (2003), Kozak (2004), Sharma and Zhang (2004) and Sharma and Parton (2009). Table 2 shows the selected taper models along with the source of each model. In all cases, the dependent variable was the diameter along the stem (d). Some studies (e.g. Gregoire et al., 2000;de-Miguel et al., 2012) have suggested that the predictions of total or merchantable volumes are less biased when d 2 is used to fit taper models. However, we used d as the dependent variable because the prediction of height h at specific upper stem diameters is an important step in classifying stem sections by upper diameter limits and log lengths as per different industrial requirements. The model of Clark et al. (1991) requires an additional measurement of diameter at 5.3 m height. At first, these measurements were attained by linear interpolation. Afterward, they were predicted with the equation proposed by Clark et al. (1991) (Table 2). The Clark et al. (1991) Model 1 and Clark et al. (1991) Model 2 represented interpolation and prediction methods, respectively.

Model fitting
The model parameters were estimated with the MODEL procedure of SAS using the generalized nonlinear least-squares method, which allows direct modelling of the error variance for autocorrelation in the fitting process (SAS Institute Inc., 2008). Spatial correlation was expected within the observations due to hierarchical data of the study. We instituted a second-order continuous autoregressive error structure (CAR (2)) to adjust the Forestry Table 2 Tested stem taper models Max and Burkhart (1976) Clark et al. (1991) proposed following equation to predict F:

Variable Exponent Taper Models
Muhairwe (1999) Stem taper functions for white birch and costata birch  Sharma and Parton (2009)  innate autocorrelation. This specified error structure allows the practical use of a model for irregularly spaced and unbalanced data (Grégoire et al., 1995). Programming for CAR (2) structure was worked out in the MODEL procedure of SAS (SAS Institute Inc., 2008).

Model comparisons
The accuracy of diameter and volume estimates was evaluated by graphical and numerical assessments of the residuals. The volumes of the following timber assortments were calculated (1) total volume (V T ), volume from stump height to tip; (2) pulpwood volume (V P ), volume from stump height to 10 cm top diameter and (3) saw log (V S ) volume from stump height to 20 cm top diameter. Stump height was 0.3 m in this analysis. The measured diameters were used to calculate sectional volumes, which were added to obtain observed V T , V P and V S . Similarly, the predicted diameters were utilized to calculate the predicted volume assortments. The length of sections varied from 0.3 to 1 m along the stem. The volume of each section was calculated by Smalian's formula. Four standard goodness-of-fit statistics were tested: root mean square error (RMSE), fit index (FI), mean absolute bias (MAB) and mean percentage of bias (MPB). 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 models were also assessed by box plots of d residuals against position (relative heights of 5, 15, 25, up to 95 per cent).
Likewise, the total volume residuals were plotted against diameter classes. These graphs portray the domains of inadequate or acceptable predictions (Kozak and Smith, 1993).

Ranking of models
The models were also compared using the ranking method of Poudel and Cao (2013).
Where R i indicates the relative rank of a model i (i = 1, 2, 3 . . . m), S i is the goodness of fit statistics delivered by model i and S max and S min correspond to the maximum and minimum values of S i . Rank 1 represents the best model, whereas m shows the poorest model. The ranking method was applied using RMSE, FI, MAB and MPB statistics for diameter, total and partial volumes to calculate the average rank of each model. Next, the mean of average ranks was taken to determine the overall ranks of the models for the four variables.

Nonlinear mixed-effect modelling of the selected taper model
After deciding the best model, it was refitted using nonlinear mixed-effects modelling method to account for within and between tree variations on stem profile. We used the NLMIXED procedure in SAS (SAS Institute Inc., 2008) to estimate the fixed and random parameters. Different combinations of random parameters were tested, which produced several mixed models. Mixed models were compared by Akaike's Information Criterion (AIC) (Akaike, 1974), Schwarz's Bayesian Information Criterion (BIC) (Schwarz, 1978) and twice the negative log-likelihood (−2Ln (L)).
Where n is the number of observations, k is the number of parameters and L is the maximum likelihood value.

Results
The initial fitting of the models without the error structure resulted in strong autocorrelation. As an example, the autocorrelation observed in white birch for d predictions with the Clark et al. (1991) Model 1 is shown in Figure 2. This correlation trend disappeared when a second-order CAR (2) was added in the model fit. Tables 3 and 4 show the parameter estimates, and Table 5 highlights the fit-statistics of the models. Above 98 per cent of the total variance of d was explained by the models for white birch, and it was almost 95 per cent for costata birch (Table 5). The Clark et al. (1991) Model 1 showed the best results for both species (RMSE: 0.875-1.952 cm, MAB: 0.595-1.313 cm and MPB: 4.763-5.045 per cent). The model of Lee et al. (2003) indicated the largest variability and bias (MAB: 0.731 cm, MPB: 5.847 per cent for white birch; MAB: 1.574 cm, MPB: 6.048 per cent for costata birch). The models of Kozak (2004), Muhairwe (1999) and Bi (2000) provided good results in terms of error and bias for white birch, whereas the models of Max and Burkhart (1976), Clark et al. (1991) Model 2 and Kozak (2004) performed better for costata birch. Although MAB for costata birch was higher than white birch, the MPB was almost similar (5-6 per cent) for both species. The Clark et al. (1991) Model 1 produced better results than the Clark et al. (1991) Model 2. The latter increased the bias by almost 6 per cent for costata birch and 12 per cent for white birch. When diameter measurements at 5.3 m were not available, the models of Kozak (2004) and Max and Burkhart (1976) best predicted the diameter for white birch and costata birch, respectively.
The Clark et al. (1991) Model 1 showed the lowest values of RMSE (0.007-0.049 m 3 ), MAB (0.004-0.036 m 3 ) and MPB (1.99-4.05 per cent) in predicting V T , V P and V S of both species (Tables 6-8). The second best models were Kozak (2004) and Bi (2000) for all volume assortments of white birch. However, for costata birch, the next best models varied across the volume assortments. In estimating V T and V P , the models of Kozak (2004), Max and Burkhart (1976), Sharma and Zhang (2004) and Lee et al. (2003) showed good results with marginal differences. Nevertheless, the model of Max and Burkhart (1976) delivered the lowest RMSEs for these attributes. In predicting V S , the models  Stem taper functions for white birch and costata birch  of Lee et al. (2003) and Sharma and Parton (2009) performed better by nominal values. In general, the models had a 2-5 per cent bias for the volume attributes. The models of Sharma and Zhang (2004) and Parton (2009), andFang et al. (2000) showed the highest RMSEs for white birch and costata birch, respectively. According to the average rank of diameter and volume estimates, Clark et al. (1991) Model 1 showed the lowest rank for both species (Table 9). When diameter measurements at 5.3 m were not available, the best models were Kozak (2004) and Bi (2000) for white birch, whereas the Kozak (2004) and Max and Burkhart (1976) were the leading models for costata birch with a similar rank. Results also indicated that the Clark et al. (1991) Model 1 was significantly better than Clark et al. (1991) Model 2.
The box plots of d residuals reflected that the distribution of error was not similar across the relative height classes (Figures 3  and 5). The prediction errors were relatively larger for costata birch. The Clark et al. (1991) Model 1 showed the best predictions for the basal log and middle stem sections. The models of Kozak (2004), Max and Burkhart (1976), Clark et al. (1991) Model 2, Fang et al. (2000) and Bi (2000) delivered reasonable estimates for the lower and middle stem. However, the models of Max and Burkhart (1976) and Clark et al. (1991) Model 2 were slightly biased for the middle sections (25-55 per cent), whereas Kozak (2004) and Bi (2000) models showed a minor distortion for the basal log (<10 per cent). The rest of the models had problems in their predictions as a whole.
The plots of total volume residuals indicated larger errors for bigger trees (Figures 4 and 6). The Clark et al. (1991) Model 1 predicted the total volume more accurately except for the largest diameter class, where the Kozak (2004) model performed better. The models of Max and Burkhart (1976), Clark et al. (1991) Model 2 and Fang et al. (2000) slightly underestimated smaller trees (10-20 cm) of white birch. This deviation was extended to all diameter classes of costata birch. The models of Muhairwe (1999), Bi (2000), Lee et al. (2003), Sharma and Zhang (2004) and Sharma and Parton (2009) overestimated total volume for all diameter classes. The taper simulation for a small and large tree of white birch and costata birch indicated more taper variation in the former. The species differed greatly in the middle and upper sections of the stem. The predictions of stem profiles using Clark et al. (1991) Model 1 are shown in Figure 7.    Through the above model comparisons, the Clark et al. (1991) Model 1 was the best model owing to its accuracy in all facets of diameter and volume estimates. This model was changed to a nonlinear mixed-effects model where random components were added to some of the fixed-effects parameters to obtain treespecific parameter estimates for trees in the study. We fitted Clark Stem taper functions for white birch and costata birch  (1991) Model 1 with possible combinations of fixed-and random-effects parameters. 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, which produced a total of 21 mixed models. Table 10 shows the evaluation statistics of the diameter along the stem for the model with the best one and two randomeffects parameters as well as fixed-effects models. The mixed models with two random parameters, u 2 and u 6 , showed the lowest values of AIC, BIC and -2Ln (L). The fitting statistics of mixed models were significantly better than the fixed-effects models. Although introducing the random-effects can reduce the cor-related errors, the existence of autocorrelation cannot be ruled out (Garber and Maguire, 2003;Trincado and Burkhart, 2006). Therefore, CAR (1) was introduced into the mixed models, which further improved the fit statistics. Table 11 shows the evaluation statistics for the model with fixed-and mixed-effects models and fixed parameters of mixed-effects models for volume (m 3 ) estimates. The comparison of fixed-effects model, fixed parameters of mixed-effects model and mixed-effects model was carried out. The mixed models showed the lowest values of RMSE, MAB and MPB for both of the species. Table 12 lists the estimated fixed parameters, correlation parameter and variance components of the mixed models for white birch and costata birch.
The adaptation of Clark et al. (1991) Model 1 was:  where u 2 and u 6 are the random parameters and all other variables as defined earlier.

Discussion
Several taper studies have presented different taper models for birch species in the world. For example, Max and Burkhart (1976) for B. alleghaniensis and B. lenta in Virginia (Martin, 1981); Kozak (1988) for Saskatchewan B. papyrifera (Gál and Bella, 1994); Kozak (2004) for B. papyrifera, B. alleghaniensis, B. lenta and B. populifolia in Acadian region, USA (Weiskittel et al., 2011) and Ung et al. (2013) for B. papyrifera and B. alleghaniensis across Canada. Recently, Shahzad et al. (2019Shahzad et al. ( , 2020 suggested the models of Fang et al. (2000) and Max and Burkhart (1976) (Burger and Shidong, 1988;Xiao et al., 2002). The accuracy of a taper model depends on the particular tree species, stem-form and stand characteristics (Muhairwe et al., 1994). It is not practicable to use a single taper equation uniformly for a vast range of conditions (Newnham, 1988). This study evaluated 10 well-known taper models for estimating diameter, total volume and merchantable volume (pulpwood and sawlog) of white birch and costata birch. Although several studies have evaluated these models, previous research has mainly focussed on conifer species. For broadleaved tree species, the model of Clark et al. (1991) has only been tested for yellow poplar (Liriodendron tulipifera L.) in West Virginia, USA (Jiang et al., 2005).
In our initial evaluation using fixed-effects models, the Clark et al. (1991) Model 1 provided the most accurate results across Stem taper functions for white birch and costata birch σ 2 , residual variance; ρ, correlation parameter for the CAR (1) error structure; var (u 2 ) and var (u 6 ), variances for the random effects corresponding to fixed parameters b 2 and b 6 ; cov (u 2, u 6 ), covariances between pairs of random effects.
the datasets, which was refitted to develop mixed-effects models. In estimating diameter, it had RMSEs that were 3.71 and 2.91 per cent lower than the next best models, Kozak (2004) for white birch and Max and Burkhart (1976) for costata birch, respectively. In estimating V T , V P and V S , the Clark et al. (1991) Model 1 reduced the RMSE by 6.32-22.52 per cent for white birch and 8.08-13.24 per cent for costata birch, when compared with the next models. For the residuals' plots, the Clark et al. (1991) Model 1 displayed the highest accuracy in terms of mean prediction error, medians' distribution and narrowness of interquartile ranges (Figures 3-6). The models of Kozak (2004), Fang et al. (2000 and Max and Burkhart (1976) have displayed a similar distribution of d residuals for pedunculate oak (Quercus robur L.) and Scots pine (P. sylvestris) in Spain and Himalayan birch (B. alnoides) in South China, respectively (Barrio Anta et al., 2007;Crecente-Campo et al., 2009;Tang et al., 2017). Contrary to the corresponding plots in Schröder et al. (2014), the models of Kozak (2004), Max and Burkhart (1976) and Bi (2000) were more precise in this analysis. The prediction accuracy of the models gradually decreased with increasing the diameter classes. Diéguez-Aranda et al. (2006) and Shahzad et al. (2019) observed similar errors for bigger trees. Schröder et al. (2014) attributed this deflection to the difference in site and competition conditions that affect individual trees.
The model of Clark et al. (1991) is comprised of Schlaegel's form-class model and Max and Burkhart's segmented model. Schlaegel's model contains Girard's form class height (5.3 m above ground), which enables a single species model to predict taper formation accurately in different geographic or physiographic regions. Clark et al. (1991) Clark et al. (1991) model, the locally calibrated models were more precise in volume estimates. Li and Weiskittel (2010) proposed the Clark et al. (1991) Model 1 for volume estimates of balsam fir (Abies balsamea (L.) Mill.), red spruce (Picea rubens Sarg.) and white pine (Pinus strobus L.) in North America. However, in that study, Kozak (2004) and Bi (2000) superseded Clark et al. (1991) Model 1 in diameter estimates. Moreover, the models of Fang et al. (2000) and Kozak (2004) were significantly superior to Max and Burkhart (1976) in predicting diameter and total volume. In our analysis, the performance of the Max and Burkhart (1976) model was better than Fang et al. (2000) and similar to Kozak (2004) for diameter and volume estimates of costata birch. We found that Clark et al. (1991) Model 1 produced lower RMSEs than Clark et al. (1991) Model 2 (by 2.88-7.13 per cent for diameter and 6.25-16.66 per cent for total volume). Li and Weiskittel (2010) recommended Forestry other well-behaved models when diameter measurements at 5.3 m height are not available. In this analysis, the models of Kozak (2004) and Max and Burkhart (1976) were superior to other models for white birch and costata birch, respectively. Previously, the model of Max and Burkhart (1976) has shown excellent results for several Appalachian hardwoods and B. alnoides in South China (Martin, 1981;Tang et al., 2017). Similarly, the Kozak (2004) model was suggested for hardwood species in the northeastern US, and it performed very well for Castanea sativa Mill. in northwest Spain (Weiskittel et al., 2011;Menéndez-Miguélez et al., 2014).
Similar to our results, Figueiredo-Filho et al. (1996) found that Clark et al. (1991) Model 1 best predicted the diameter and total or merchantable volume of loblolly pine (Pinus taeda L.) in Brazil. They also observed that the Max and Burkhart (1976) model demonstrated quite reasonable predictions for all variables. Unlike Özcelik and Brooks (2012), the model of Max and Burkhart (1976) performed better than Clark et al. (1991) Model 2 for costata birch in our analysis. Sakici et al. (2008) assigned a similar rank to Clark et al. (1991) Model 1 and Kozak (2004) for diameter estimates of A. nordmanniana in Turkey, which was not the same in this analysis. In agreement with Doyog et al. (2017), the model of Kozak (2004) was superior to the Clark et al. (1991) Model 2 in this study.
Costata birch trees indicated a different trend from white birch for diameter and volume predictions in this study (e.g. higher variance and greater bias in the models). This might be attributed to the larger size of trees available in the costata birch dataset than white birch (Figure 1, Table 1). This variation was also reflected in the taper simulations of the species (Figure 7). The species had an identical taper formation for the basal log for given tree size, but white birch showed more taper than costata birch in the middle and upper stem sections. However, the Clark et al. (1991) Model 1 best predicted the diameter, total volume and merchantable volume of both species.
We selected the Clark et al. (1991) Model 1 for developing the mixed models due to its consistent superiority across the datasets. However, the future user who likes to use the models of Kozak (2004) or Max and Burkhart (1976), the parameters for fixed-effects are given in Tables 3 and 4. Fixed-effect modelling, even with biased parameters, can provide good predictions (de-Miguel et al., , 2013.

Tree level mixed-effects models
It is well known that the taper model fitted by non-linear mixedeffects modelling approach can improve the goodness-of-fit statistics compared with ordinary least squares. However, for the prediction purpose, many authors suggest using the fixedeffects models in the absence of calibration data (de-Miguel et al., 2013;Arias-Rodil et al., 2015). The fixed-effects models are more accurate in the predictions when random parameters of the mixed models are supposed to be zero, and additional measurements are not available for model calibration (Pukkala et al., 2009;Shater et al., 2011;Guzmán et al., 2012). de-Miguel et al. (2013) also advised reporting both fixed-and mixed-effect forms of a model since calibration may be a feasible option in some cases. Accordingly, the mixed-effects modelling of the Clark et al. (1991) Model 1 was carried out. Mixed-effects models would be used where additional upper stem diameter measurements are available for calibration. Table 10 shows that the addition of one random parameter improved the fitting. However, the models expanded with both u 2 and u 6 provided the best statistics. The addition of random parameters u 2 and u 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 (Clark et al., 1991). The adapted Clark et al. (1991) Model 1 with two random parameters allows for the variation in both segments between trees.
The expansion of the models with random effects u 2 and u 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) ( Table 2). The random-effects affected all parameters b 1 -b 4 , but the parameter b 2 showed the best statistics amongst different combinations. The variation is usually minimum in the lower stem (see, for example, Clark et al., 1991, Figure 3/ Table 4). The parameter b 2 explains the basal log and compensates for the slightly tapered lower stem. The parameter b 6 captures the variation in the middle stem and joins b 5 to control the upper stem. With a higher value, the parameter b 6 has a major impact on the model, and the parameter b 5 contributes a little. Therefore, the parameter b 6 can explain the tree-level variation in both sections. Since the Clark et al. (1991) Model uses the Max and Burkhart (1976) type for these sections, a parallel can be drawn with the findings of Leites and Robinson (2004). While fitting the Max and Burkhart (1976) model with mixedeffects, they used the same terms to describe the middle and upper stem sections of loblolly pine (P. taeda) in Uruguay.
Although improved accuracy in the mixed models is appealing, the improvement in model accuracy for individual trees depends on the calibration data that should be as accurate as modelling data. Substantial biases or inaccuracies could lead to weaker estimates than those received from the fixed-effects model (Westfall and Scott, 2010).

Conclusions
This study presents an initial attempt to develop stem taper models using fixed-and mixed-effects modelling for white birch and costata birch in northeast China. The Clark et al. (1991) Model 1 was superior to other taper models in predicting the diameter at any height and total or partial volumes across the datasets. As an additional benefit, this model is compatible, which can be integrated to estimate merchantable and total volume. The Clark et al. (1991) Model 1 was also presented as a mixed model. The addition of random effects improved the precision of the model. However, mixed-effects models would be used where additional upper stem diameter measurements are available for calibration. When diameter measurements at 5.3 m are not available, the models of Kozak (2004) and Max and Burkhart (1976) demonstrated quite reasonable results for white birch and costata birch, respectively.
White birch and costata birch are widely distributed in northeast China, which has significant geographic differences. Future analysis with a sample of larger representation could account for