Dahurian larch (Larix gmelinii Rupr.), Korean spruce (Picea koraiensis Nakai), and Manchurian fir (Abies nephrolepis Maxim) are valuable tree species of northeast China. Management tools are required for these species to support the industrial and ecological development of forest ecosystems of China. This study aimed to develop taper equations for these species. Five well-known taper equations from two groups (segmented and variable form taper models) were evaluated using the fixed- and mixed-effect modeling. The models were independently fitted utilizing diameter and height datasets comprising 473 trees. A third-order continuous autoregressive error structure (CAR (3)) was used to account for autocorrelation. The model of Clark et al. (1991. Stem profile equations for southern tree species. Asheville, NC: USDA Forest Service. Research Paper, SE-282. doi:10.2737/SE-RP-282) was superior to other models for estimating diameter, merchantable volume, and stem volume when upper stem diameter at 5.3 m was available or predicted. After model comparison, the best Clark et al. (1991) model was refitted as a nonlinear mixed-effects model. This study also demonstrated that the fixed-effect model of Clark et al. (1991) was more accurate than Kozak (2004) mixed model when the later was calibrated for upper stem diameter at 5.3 m with validation data.

Key message: The non-linear seemingly unrelated regression mixed-effects model (NSURMEM) and generalized additive model (GAM) were applied for the first time in crown width (CW) additive models of larch (Larix gmelinii Rupr.), birch (Betula platyphylla Suk.), and poplar (Populus davidiana Dode). The crown radii in four directions (CR) exhibited different growth trends and responded differently to tree size and competition variables. In the absence of calibration, GAM was more accurate than NSURMEM for CR and CW predictions. Context: Crown radii in four directions (CR) and crown width (CW) are fundamental indicators used to describe tree crowns. The complexity of the CR growth in four directions of different tree species in natural forests is often ignored. There is logical additivity among CR and CW that is also often overlooked. Furthermore, the existing methods applied to CW additive models have some drawbacks. Aims: We aim to: (i) evaluate the utility of two new methods in developing CW additive models for larch (Larix gmelinii Rupr.), birch (Betula platyphylla Suk.), and poplar (Populus davidiana Dode) in natural secondary forests of Northeastern China; and (ii) explore the growth patterns of CR in four directions to gain important ecological insights. Methods: The non-linear seemingly unrelated regression mixed-effects model (NSURMEM) and generalized additive model (GAM) were used to develop CW additive models and to explore crown growth patterns. The predictive ability of the additive models was evaluated using leave-one-plot-out cross-validation (LOOCV). Results: At a fair level without calibration, GAM provided slightly better results than NSURMEM. The response of the four CR to tree size and competition variables is different and may be non-uniform due to complex stand conditions and tree growth strategies. Conclusion: The newly provided methods applied to additive models are available for external datasets. GAM is recommended in the absence of calibration. This study has important implications for the understanding of natural forest dynamics and decision-making for critical stand management.

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.

Accurate prediction of stem diameter is an important prerequisite of forest management. In this study, an appropriate stem taper function was developed for upper stem diameter estimation of white birch (Betula platyphylla Sukaczev) in ten sub-regions of the Daxing’an Mountains, northeast China. Three commonly used taper functions were assessed using a diameter and height dataset comprising 1344 trees. A first-order continuous-time error structure accounted for the inherent autocorrelation. The segmented model of Max and Burkhart (For Sci 22:283–289, 1976. https://doi.org/10.1093/forestscience/22.3.283) and the variable exponent taper function of Kozak (For Chron 80:507–515, 2004. https://doi.org/10.5558/tfc80507-4) described the data accurately. Owing to its lower multicollinearity, the Max and Burkhart (1976) model is recommended for diameter estimation at specific heights along the stem for the ten sub-regions. After comparison, the Max and Burkhart (1976) model was refitted using nonlinear mixed-effects techniques. Mixed-effects models would be used only when additional upper stem diameter measurements are available for calibration. Differences in region-specific taper functions were indicated by the method of the non-linear extra sum of squares. Therefore, the particular taper function should be adjusted accordingly for each sub-region in the Daxing’an Mountains.

The applicability of height to crown base (HCB) appears to be particularly broad given the important impact on explaining tree growth dynamics. Given the high cost of measuring HCB, accurate prediction with a few trees is popular in forestry practice. In this study, a fixed-effects model (FEM), a mixed-effects model (MEM), and quantile regressions (QR) were adopted to predict HCB using data from natural secondary forests of Dahurian larch (Larix gmelinii Rupr.) in Northeastern China. Corresponding calibration techniques were applied to trees with different sample designs (random selection, the thickest tree selection, the intermediate tree selection, and the thinnest tree selection) and sample sizes (1–10 trees per plot). The results showed that all models achieved accurate HCB predictions. The QR calibration technique of 3-quantiles achieved simple and accurate consequences compared to 5-quantiles and 9-quantiles. MEM displayed the most robust and superior statistics. The selection of the two thickest trees was recommended for the MEM. For FEM and QR, the sample size should be exceeded by 5. Generally, a combination of MEM and sampling two thickest trees per plot to predict HCB is recommended as a win–win solution that neither sacrifices model prediction accuracy nor minimizes the required measurement cost.