Parameterization of height–diameter and crown radius–diameter relationships across the globe

Abstract

The tree height–diameter at breast height (H–DBH) and crown radius–DBH (CR–DBH) relationships are key for forest carbon/biomass estimation, parameterization in vegetation models and vegetation–atmosphere interactions. Although the H–DBH relationship has been widely investigated on site or regional scales, and a few of studies have involved CR–DBH relationships based on plot-level data, few studies have quantitatively verified the universality of these two relationships on a global scale. This study evaluated the ability of 29 functions to fit the H–DBH and CR–DBH relationships for six different plant functional types (PFTs) on a global scale, based on a global plant trait database. Results showed that most functions were able to capture the H–DBH relationship for tropical PFTs and boreal needleleaf trees relatively accurately, but slightly less for temperate PFTs and boreal broadleaf trees (BB). For boreal PFTs, the S-shaped Logistic function fitted the H–DBH relationship best, while for temperate PFTs the Chapman–Richards function performed well. For tropical needleleaf trees, the fractional function of DBH satisfactorily captured the H–DBH relationship, while for tropical broadleaf trees, the Weibull function and a composite function of fractions were the best choices. For CR–DBH, the fitting capabilities of all the functions were comparable for all PFTs except BB. The Logistic function performed best for two boreal PFTs and temperate broadleaf trees, but for temperate needleleaf trees and two tropical PFTs, some exponential functions demonstrated higher skill. This work provides valuable information for parameterization improvements in vegetation models and forest field investigations.

摘要

全球尺度上树高-胸径和树冠半径-胸径关系的参数化方案

树高-胸径(H–DBH)、树冠半径-胸径(CR–DBH)函数关系不仅是森林碳库/植被生物量估算的关键，同时也是植被模式参数化方案的重要组成部分，对陆-气相互作用的模拟起着至关重要的作用。目前，已有大量工作研究了站点或区域尺度的H–DBH函数关系，也有少量工作基于站点观测数据探讨了CR–DBH函数关系，但鲜有研究在全球尺度上定量研究这些函数关系的普适性。因此，本文基于全球植物性状数据库，率先在全球尺度上评估了29种函数对6种不同植物功能型 (Plant Functional Types, PFTs)的H–DBH和CR–DBH关系的拟合能力。研究结果显示，大多数函数能够较准确地刻画热带PFTs和寒带针叶林的H–DBH关系，但对温带PFTs和寒带阔叶林的拟合稍差。Logistic函数最适合拟合寒带PFTs的H–DBH关系，而对于温带PFTs而言，Chapman–Richards函数表现较好。分数函数能够较好地刻画热带针叶林H-DBH关系，而Weibull函数和一些复合函数则是热带阔叶林的最佳选择。对于大多数PFTs(寒带阔叶林除外)而言，所有函数对CR–DBH关系的拟合能力相当，其中，Logistic函数对两种寒带PFTs和温带阔叶林的CR–DBH关系拟合最佳，而一些指数函数对温带针叶林和两种热带PFTs的拟合效果更好。本研究为植被模式参数化方案的改进和森林实地调查研究提供了宝贵的信息。

INTRODUCTION

Tree height (H), diameter at breast height (DBH) and crown radius (CR) are three basic physical quantities that describe the growth and size of individual trees (Larjavaara 2021). They are closely related to the biophysical and biochemical processes of ecosystems. H governs light access (Falster and Westoby 2003) and seed dispersal distance (Thomson et al. 2011), while DBH influences water transport and mechanical support within the plant, and also affects leaf biomass. The relationship between these variables reflects the trade-off in the ‘growth versus survival’ life strategy. Additionally, CR exhibits a strong relationship with leaf area (Kenefic and Seymour 1999), directly impacting photosynthesis and evapotranspiration, as well as forest microclimate dynamics, which drive plant responses to warming (Zellweger et al. 2020).

How to describe the relationships of H–DBH and CR–DBH is of significant concern to ecosystem modelers who develop mathematical representations or parameterization schemes in Earth System Models (ESMs) or Dynamic Global Vegetation Models (DGVMs) (Ma et al. 2022; Suwa 2013; Weng et al. 2015; Zeng et al. 2014). This is because an inadequate description of the morphology of individual trees can lead to simulation biases in H (Dai et al. 2020), leaf area index (LAI) (Song et al. 2021), carbon fluxes (Toda et al. 2023), vegetation biomass (Song et al. 2017), fractional coverage distributions and more. Such biases can further propagate into simulation biases of canopy precipitation interception (Klamerus-Iwan et al. 2020), radiative transfer (Yuan et al. 2014), evapotranspiration, surface albedo (Hartley et al. 2017) and even temperature and precipitation (Dong et al. 2016; Huang et al. 2018) in vegetation–atmosphere interaction. Furthermore, due to the dependence of stem volume on H and DBH, and the effects of tree canopy on carbon and water cycles, the H–DBH and CR–DBH relationships are also crucial for foresters when estimating forest carbon/biomass (Ni et al. 2017; Sato et al. 2015), as well as ecologists when studying the climate and environment (Ryan 2002).

Up to now, studies of observed tree H–DBH relationships have mostly applied nonlinear concave functions (e.g. Alvarez-Buylla and Martinez-Ramos 1992; Jucker et al. 2022; Kooyman and Westoby 2009; Lines et al. 2012; Rasmussen et al. 2011) or S-type functions (e.g. Logistic function, Weibull function and Schnute function) (Huang and Trrus 1992; Huang et al. 2000). It has been shown that power functions and exponential functions are applicable in describing the H–DBH relationship for tropical forests (Kooyman and Westoby 2009; Osunkoya et al. 2007), while S-type functions are more appropriate for boreal forests (Huang and Trrus 1992). For temperate forests, both power functions (Zhao et al. 2021) and S-type functions (Ahmadi et al. 2013) have been adopted. However, in most of current DGVMs, power functions formed of $H=a×DBHb$ (e.g. Maignan et al. 2011; Moorcroft et al. 2001; Zeng et al. 2014) and fractional function $H=(1/(a×DBH)+1/b)−1$ (Suwa 2013) have been widely used to describe tree individual growth in globe, without distinction among different plant functional types (PFTs). Unified simple function forms and parameters across the globe simplifies parameterization in DGVMs, but it may make large biases in vegetation structure and biomass.

In contrast, there have been relatively few studies on the CR–DBH relationship, and most that have been conducted have focused on coniferous forests (Sileshi et al. 2014). For example, ln(CR)–ln(DBH) (Hasenauer 1999; Meng et al. 2007) and CR–ln(DBH) (Novotný et al. 2020) linear models have been used for needleleaf evergreen trees, while a CR–DBH linear function was used to describe the morphology of needleleaf deciduous trees (Gilmore 2001). Sileshi et al. (2014) compared four power functions between the crown diameter (CD) and DBH for tropical trees and found that the CD–DBH linear function (model exponent = 1) had the smallest Akaike information criterion value.

Some studies have investigated the morphological variances of different tree species via parameter selections, based on typically adopted existing models without identifying their appropriateness. Actually, the choice of different fitting functions often reflects the diverse ways in which trees allocate photosynthetic products, and their different mathematical properties also imply different hypotheses. Some studies have suggested that functions with asymptotic behavior can effectively represent the hydraulic constraint hypothesis (Wood et al. 2015; Zhao et al. 2021). However, Bontemps and Duplat (2012) argued that asymptotic functions tend to underestimate the inherent growth rate of the tree, and proposed that functions with non-asymptotic behavior could better describe the growth process of the tree. Hence, the selection of the best fitting functions should be a preliminary step before further analysis (Huang et al. 2000; Temesgen et al. 2014).

Existing studies have derived morphological functions based on site or regional observations with a limited number of samples and species, and universality of these models has not yet been verified across different ecoregions, or on a global scale. Hence, applying these morphological functions in ESMs/DGVMs may lead to significant biases in predicted tree traits (Huang et al. 2000) and global vegetation distributions (Song et al. 2021).

Therefore, in this study, the optimum fitting functions for the H–DBH and CR–DBH relationships for different PFTs were determined through validation and comparison among 29 functions, based on a global plant trait dataset. Furthermore, their mathematical properties were also thoroughly examined. The aim in carrying out this work was to provide a valuable foundation for improving parameterization in vegetation models, and to offer insights for forest field investigations.

DATA AND METHODS

Data

The data used in this study were obtained from Jucker et al. (2022). They encompass a collection of 498 839 trees sampled from 61 856 unique sites around the world. The data cover all major forest and non-forest biomes, including data on 5163 species from 1453 genera and 187 plant families. The dataset mainly includes information on whether samples are angiosperms or gymnosperms, along with their family/genus/species names, the latitude and longitude of the sites and the DBH (cm), H (m) and CR (m) of the trees.

Because not all the samples have a complete set of information, in this study, a total of 418 643 observations were included, encompassing detailed information as mentioned above. Then, they were classified into needleleaf and broadleaf trees based on their species names, and three climate zones with the criteria of tropical zones 23.5° S–23.5° N, temperate zones 23.5° S–66.5° S or 23.5° N–66.5° N and boreal zones 66.5° S–90° S or 66.5° N–90° N. At last, the observational data were divided into six PFTs based on leaf traits and climate zones (Table 1). This gave a subset of 102 886 observations of needleleaf trees, with 1658 from boreal regions, 100 312 from temperate regions and 916 from tropical regions. For broadleaf PFTs, the dataset contained 315 757 observations, with 265 from boreal regions, 161 963 from temperate regions and 153 529 from tropical regions.

Methods

Twenty-nine different functions were used to model the H–DBH and CR–DBH relationships (Table 2). These functions encompassed linear functions (Func.24), logarithmic functions (Func.25), fractional functions (e.g. Func.17, 19–22), power functions (e.g. Func.1, 28), exponential functions (e.g. Func.2–4, 26–27) and some composite functions (e.g. Func.13–15). A few of these functions were from Huang et al. (2000) and Feldpausch et al. (2011). The coefficient of determination (R²), root-mean-squared error (RMSE) were used as skill scores to assess the goodness-of-fit. If both R² and RMSE were close (R² and RMSE differences less than 0.001), functions with less parameters won. In addition, in most countries, the height of 1.3 m above the ground is defined as DBH. So, many popular functions (such as Func.1–22) have the constant term C₀ = 1.3 which makes the left-hand value of functions no less than 1.3. Therefore, when fitting Func.1–22 for H–DBH relationship, data with H less than 1.3 m were excluded.

RESULTS

Regression between H and DBH

Twenty-nine functions (Table 2) were utilized to model the H–DBH relationship. Compared to broadleaf trees, the prediction skill of these functions was generally higher for needleleaf trees in each climatic zone, with the average R² ranging from 0.47 to 0.72 for needleleaf trees, and from 0.36 to 0.66 for broadleaf trees. As evidenced by the high R² and low RMSE, most of these functions could capture the H–DBH relationship relatively accurately for boreal PFTs (average R² of 0.68 and 0.41, and RMSE of 2.48 and 2.47 m, respectively) and tropical PFTs (average R² of 0.72 and 0.66, and RMSE of 4.43 and 4.84 m, respectively), but slightly less so in temperate zones (average R² of 0.47 and 0.36, and RMSE of 5.72 and 7.07 m, respectively) (Fig. 1; Supplementary Table S1).

For boreal needleleaf trees (BN), obvious differences existed in the fitting capabilities among different functions, among which Func.10 fitted best (R² = 0.76, RMSE = 2.20 m), closely followed by Func.5 and Func.12 (R² = 0.75, RMSE = 2.22 and 2.21 m, respectively) (Fig. 1a and b). For boreal broadleaf trees (BB), probably due to its lower data availability, different functions had much closer R² values, wherein the best fitting functions were Func.9, Func.13 and Func.22 (R² = 0.43, RMSE = 2.42 m) (Fig. 1c and d), but with no obvious differences (Fig. 2b). Furthermore, there were many other functions with equivalent fitting ability, such as Func.3, Func.10, Func.16 and Func.18. Notably, the best fitting functions for boreal regions were all S-shaped, consistent with the previous conclusion of Huang and Trrus (1992) that S-type functions are suitable for describing the H–DBH relationship for boreal trees. For the two PFTs in temperate regions, Func.13 performed best, achieving R² values of 0.50 and 0.40, along with RMSE values of 5.54 and 6.87 m, respectively. For temperate needleleaf trees (MN), Func.6 was comparable to Func.13, with similar R² and RMSE (Fig. 1e–h), with slight differences for large trees (Fig. 2c). For temperate broadleaf trees (MB), Func.13 obviously underestimated the H for individuals whose actual H exceeded 35 m (Fig. 2d), which was the main reason behind the low R². For tropical regions, Func.17 was the best fitting function for tropical needleleaf trees (TN), with R² of 0.77 and RMSE of 3.89 m (Figs 1i, j and 2e); while for tropical broadleaf trees (TB), Func.12 and Func.22 were comparable with R² of 0.71 and RMSE of 4.52 m (Figs 1k, l and 2f).

In addition to the best fitting functions for each category, Func.18 also had great fitting capability for all PFTs, with R² values of approximately 0.75 and 0.43 for boreal PFTs, 0.50 and 0.40 for temperate PFTs and 0.77 and 0.71 for tropical PFTs, respectively. Meanwhile, Func.12 and Func.16 were good candidates for needleleaf and broadleaf trees, respectively.

As shown in Fig. 1, there were several functions with satisfactory fitting capabilities, with very similar R² and RMSE values to the best fitting function for some PFTs. Usually, different functional forms may have different mathematical properties, such as convexity and asymptotic behavior, which reflect different individual growth processes of trees. For example, when H–DBH relationship is represented by convex functions, it will be indicated that the growth rate of tree height is first faster and then slower with DBH. But when S-shaped functions are used, the growth rate of tree height with DBH shows a slow–fast–slow process. Thus, to investigate whether existing obvious prediction differences among different functions with similar R² and RMSE, the relative predictive biases RMSE/H_obs were compared among functions with R² difference less than 0.001 (Fig. 3). For BN, Func.10 had obvious advantage (higher R²) over other functions, so only its RMSE/H_obs was shown for BN. Overall, the tendency between the relative bias and observed H was similar among the selected functions. For boreal trees, individuals with observed H of approximately 6–7 m had the highest relative biases, which was above 20% (Fig. 3a and b); while for temperate trees, the relative biases of individuals with observed H of 35–40 and 15–20 m were around 89.68%–105.47% and 145.70%–175.13%, respectively—several times higher than small and large individuals (Fig. 3c and d). Both types of tropical trees had their highest relative biases for individuals with observed H of approximately below 10 m; however, for large individuals, the relative bias decreased to below 20% for TN, but reached 40% for TB (Fig. 3e and f). On the other hand, there were significant predictive differences among different functions for individuals with relatively small or large sizes. For example, the relative biases from different functions ranged from 20.86% to 28.74% for individuals with observed H of approximately 7 m for BB, while it varied from 6.69% to 37.18% for individuals above 80 m for MB. Furthermore, the advantages of different functions were reflected in different size classes. Taking BB as an example, Func.10 had the smallest relative bias for small individuals, while Func.22 and Func.23 held the advantage for relatively tall individuals. Similar phenomena occurred between middle and large size classes for MN, MB and TB. Thus, it is evident that choosing different functions may result in significant differences in the predicted H for small and large trees, even when these functions have similar R² and RMSE values.

Regression between CR and DBH

The fitting capabilities of 27 functions (Func.1 and Func.28 were the same for CR–DBH, so did Func.2 and Func.26) were comparable for all the different PFTs except BB, with average R² values ranging from 0.60 to 0.68, and RMSE ranging from 0.42 to 1.07 m (Fig. 4; Supplementary Table S2). Similar to the H–DBH results, the prediction skill of these functions was higher for needleleaf trees than for broadleaf trees for each climatic zone. For boreal trees, the S-shaped Func.10 performed best (R² = 0.73 and 0.47, RMSE = 0.39 and 0.38 m, respectively) (Figs 4a–d and 5a, b). For temperate trees, Func.8 and Func.10 proved to be the best fitting functions for MN and MB, yielding R² values of 0.65 and 0.67, with corresponding RMSE values of 0.60 and 0.89 m, respectively (Figs 4e–h and 5c, d). For tropical trees, there were two best options for TN—namely, Func.3 and Func.16—yielding an average R² of 0.69 and RMSE of 0.85 m; while for TB, Func.8 had the highest skill scores (R² = 0.67, RMSE = 0.98 m) (Figs 4i–l and 5e, f).

Besides the best fitting functions for each PFT, Func.6, Func.13, Func.15 and Func.18 displayed great fitting capabilities for all PFTs except BN, for which the best fitting function had obvious advantages. Furthermore, aside from BN and MB, Func.28 was also a good choice for other PFTs.

Similar to the approach for the H–DBH relationship, the relative predictive biases among functions with similar R² and RMSE for the CR–DBH relationship of each PFT were also compared (Fig. 6). Generally, the relative biases of CR were large for individuals with small CR. For BB, MN and TN, significant predictive differences among selected functions appeared for individuals with relatively large CR (Fig. 6b, c and e). For BN, BB and MN, changes in relative CR biases synchronized with those of relative H biases; however, for MB, TN and TB, these changes seemed opposite when CR increased from intermediate to large values (Figs 3 and 6).

DISCUSSION AND CONCLUSIONS

The relationships among H, CR and DBH are of paramount importance for parameterizations in vegetation models and forestry estimation. While there have been numerous studies conducted at site or regional scales, until now, there has been a lack of systematic research on the universality of these fitting functions at a global scale. Therefore, this study employed 29 functions to fit H–DBH and CR–DBH relationships for six different PFTs using a comprehensive global plant trait dataset.

Results showed that different PFTs had different satisfactory morphological fitting functions. S-shaped functions were good choices for boreal trees, consistent with previous studies (e.g. Huang and Trrus 1992). The Chapman–Richard function (Func.13) was the best fitting function for the H–DBH relationship of temperate trees (MN and MB), which agrees with the outcomes reported for temperate forests by Ahmadi et al. (2013). On the other hand, although the Gompertz function (Func.9) was one of the good choices to predict the H–DBH relationship for BB, it did not have high predictive performance for needleleaf trees in all three climate regions (R² ranging from 0.46 to 0.57, RMSE ranging from 2.42 to 5.77 m). Such a finding differs from the results reported for needleleaf forests in subalpine and boreal habitats in northeast North America by Wood et al. (2015). The discrepancy is probably attributable to the use of different datasets related to regional variations.

Besides identifying the best fitting functions for each individual PFT, it was also found that the Weibull function (Func.12) and the modified Logistic function (Func.18) were also good candidates for all PFTs (except Func.12 for BB). Furthermore, the power function $H=a×DBHb$ (Func.28) also fitted the H–DBH relationship well in temperate and tropical regions, which is consistent with the results reported by Feldpausch et al. (2011) based on tropical forest trees. Some studies have reported that the quadratic polynomial function $ln(H)=a×ln(DBH)+b×(ln(DBH))2$ fits well for deciduous trees in North America (Niklas 1995), and for tropical/subtropical forests and woodland savannas (Chave et al. 2015). However, compared with $H=a×DBHb$ (Func.28, equivalent to $ln(H)=lna+b×ln(DBH)$⁠, the function $ln(H)=a×ln(DBH)+b×(ln(DBH))2$ did not obviously improve the fitting score for the H–DBH or CR–DBH for all PFTs in our dataset.

There are a limited number of studies on the CR–DBH relationship in the published literature. In the present study, besides the best fitting functions for each PFT, exponential functions (Func.3 and 8) and the Logistic function (Func.10) satisfied predictive results regarding the CR–DBH relationship. Furthermore, Func.28 was a good choice for all PFTs. For TB, the conclusion was not consistent with Sileshi et al. (2014), who reported the best fitting function to be a linear function between CR and DBH (Func.24). In addition to the data of different quantity and observation ranges, another reason leading to different optimal fitting functions may be the difference in the space or light competition intensity among neighboring individuals.

In most ecological numerical models, the power function (Func.28) is used to describe H–DBH and CR–DBH relationships, but this leads to obvious biases in tree morphology and biomass. The best fitting functions identified in this study are expected to improve tree morphology parameterizations, and then reduce model simulations biases in ecosystem structure (tree height, crown area, LAI, etc.) and biomass in DGVMs, and even improve model performance on vegetation–atmosphere interactions in ESMs. However, further improvements are still needed. First, many ecological models have considered leaf phenology (evergreen/deciduous). But Jucker’s original database provides data at the species level without information about leaf phenology. When it was aggregated at the PFT level, the phenology of the same species may differ in different regions and/or climatic zones, which might lead to inaccurate leaf phenological classification. So, more data should be further collected to take it into account. Second, this study has shown that the relative biases in predicted H and CR were high for small or medium-sized individuals. When the selected functions in this study are adopted to estimate forest carbon stocks (static target), there is no need to focus on the fitting accuracy for small individuals because a large proportion of the predictive RMSE derives from the prediction of large individuals. However, when they serve dynamic modeling, attention should be paid to predictions of small individuals. This is because large trees usually grow from small ones, and biases for small individuals not only generate biases in their resource competition, growth and mortality rates, but also further result in irrationality of ecosystem structure and succession time scales. In addition, we observed large standard deviations of H and CR for a given DBH, which suggests that other factors, such as climate, soil characteristics and individual competition likely contribute to the spatial heterogeneity in H–DBH and CR–DBH relationships. Therefore, in the next step for this line of research, the effects from abiotic factors on H–DBH and CR–DBH relationships will be further investigated to gain a more comprehensive understanding of the complexities underlying tree growth patterns and crown characteristics.

Supplementary Material

Supplementary material is available at Journal of Plant Ecology online.

Table S1: H–DBH functions.

Table S2: CR–DBH functions.

Funding

This study is supported by the National Natural Science Foundation of China (42275177) and the National Key Scientific and Technological Infrastructure project ‘Earth System Science Numerical Simulator Facility’ (EarthLab).

Data Availability

The data used in this study are from Jucker et al. (2022), and can be downloaded on Zenodo (https://doi.org/10.5281/zenodo.6637599).

Conflict of interest statement.

The authors declare that they have no conflict of interest.

REFERENCES