The Barro Colorado Nature Monument is a protected area in central Panama consisting of Barro Colorado Island (BCI) and peninsulas on the surrounding mainland (Leigh, 1999). The vegetation is tropical moist forest. Annual rainfall averages 2657 mm (years 1926 to 2017), with a 4-month dry season from approximately mid-December to mid-April (Paton, 2018). The forest dynamics plot on BCI is a 50 ha area (1000 m × 500 m) in which all tree stems with trunk diameter of at least 1 cm have been measured, mapped, tagged, and identified to species in regular censuses since the early 1980s (Hubbell, 1983; Condit, 1998; Leigh, 1999; Hubbell et al., 1999, 2005). The plot is mostly old-growth forest of 400 years or older (Piperno, 1990), with the exception of a small area of secondary forest that is ∼127–137 years old in the central part of the northern edge of the plot (Mascaro et al., 2011). The Gigante peninsula on the nearby mainland is covered by secondary forest ranging from 100 to perhaps 300 years old (Denslow and Guzman, 2000; Wright et al., 2011).

The allometric data consist of measurements of trunk diameter at 1.3 m height or above buttresses, D (cm), tree height, H (m), and crown area, C (m2). We used a compilation of seven datasets collected in the BCI 50 ha plot and one dataset from old-growth forests on the adjacent Gigante peninsula (see Table S1 in Section S1 of the Supplement for further details). The datasets cover different size classes and combine measurements made with different, albeit standard, methods. Depending on the dataset and tree size, tree heights were measured with a telescoping pole (smaller trees only), with a laser rangefinder using the sine or tangent method (Larjavaara and Muller-Landau, 2013), or from the difference between a model obtained from photogrammetry (of the ground) and that obtained from airborne lidar (only fully sun-exposed trees). Crown areas were from ground-based measurements of crown radii or from delimiting fully sun-exposed crowns in high-resolution aerial photos. We included only species with at least five individual measurements of either H or C and those that have data for the three trait covariates (see below), which resulted in a pool of 162 species, including 9884 trees for height allometries and 2425 trees for crown area allometries.

We considered three species-level covariates to assess whether functional traits can explain interspecific variability in allometric scaling: the structural trait of wood density and two demographically based indicators of shade tolerance. Species-specific wood density values (dry matter weight per unit of fresh volume; g cm-3) – technically wood specific gravity (Williamson and Wiemann, 2010) – were based on measurements taken in central Panama (Wright et al., 2010). The two shade-tolerance indicators were the rates of mean sapling diameter growth and of sapling mortality estimated by Condit et al. (2006) on BCI using the 5-year census data between 1982 and 2005. Sapling relative growth rates (% yr-1) were based on diameter increments for individuals between 10 and 49 mm in diameter at the initial census. Annual mortality rates (% yr-1) were also based on the monitoring of tagged individuals but included saplings with diameters between 10 and 99 mm (Condit et al., 2006). For both growth and mortality, the rates are weighted averages of means for each census interval, weighting by the number of records in a census interval.

We adopted a hierarchical Bayesian (HB) approach to analyze allometric relationships of tree height and crown area with trunk diameter (Dietze et al., 2008; Price et al., 2009; Iida et al., 2011). The HB approach provides several advantages over classic analytical frameworks (Cressie et al., 2009; Gelman, 2014), starting with the easy accommodation of complex data structures and process models. In the current context, the HB framework allowed us to simultaneously estimate (i) community-level allometries that best represent an average species in the community, (ii) species-specific allometries that capture interspecific variation, and (iii) general relationships of species-specific allometric parameters to functional traits. The estimation of general relationships is improved by properly weighting species-specific estimates, whereas species-level estimates for rare species benefit from borrowing strength from the community-level relationship. The latter aspect reduces the negative impact of outlying observations and allows inference for data-poor species, of which there are many in hyperdiverse ecosystems like tropical forests. Another important advantage of the HB approach is that it can easily handle nonlinear models, allowing us to extend the analysis beyond power functions (which are typically fitted through linear regressions on log-transformed data; Mascaro et al., 2014) to other functional relationships. Finally, the HB approach allowed us to assess the relationships of functional traits to tree allometries by explicitly modeling species-specific allometric parameters as functions of species traits.

The best tree height model combined a generalized Michaelis–Menten (gMM) function with species-specific parameters modeled as a linear function of sapling growth rates (Fig. 1a). At the community level, the best model for predicting tree height, H (m), from trunk diameter, D (cm), in the absence of species-level covariates was H=58.0D0.7321.8+D0.73. This equation incorporates the bias correction for the back transformation from log H based on the estimate of σν=0.181 [0.179, 0.183]90%. The parameter values with their 90 % posterior central intervals are asymptote a=57.1[54.5,60.0]90% before bias correction, exponent b=0.73[0.72,0.75]90%, and half-saturation parameter k=21.79[20.70,22.89]90%.

Individual species showed considerable variation in their height allometries; this variation was explained to a large extent by the sapling growth rate (Fig. 2). Parameters a and k declined with sapling growth rate, while b increased (Fig. 2; r2=0.42, 0.14, and 0.16 for relationships of the natural logarithm of sapling relative growth rate with a,b, and k, respectively). As a consequence, fast-growing species attain taller heights at small diameters but have shorter asymptotic heights compared with slow-growing species (Fig. 2d). The second-best model included a generalized Michaelis–Menten function and no covariates but was significantly worse in WAIC (ΔWAIC =5.6). The third-best model, which incorporated a Weibull function and interspecific variation in sapling growth rates (Table 1), produced the same results qualitatively as the best model. In general, all the saturating models (Weibull or gMM), regardless of covariate or not, had similar expectations of the log pointwise predictive density (ΔELP < 10), whereas the power function models did much worse (ΔELP > 500, ΔWAIC > 1000).

The allometric scaling of crown area with trunk diameter showed no sign of saturation, thus the power function model provided superior fits (Fig. 1b). At the community level, the best model for predicting crown area, C (m2), from trunk diameter, D (cm), in the absence of information on species-level covariates was C=0.66D1.34. This equation incorporates the bias correction for the back transformation from log C based on σν=0.549[0.536,0.563]90%. The parameter values with their 90 % posterior central intervals are coefficient a=0.57[0.50,0.63]90% before bias correction and exponent b=1.34[1.31-1.38]90%. This model does not have an asymptote. For the maximum trunk diameter in our dataset, Dmax=250 cm, we would expect a crown area close to Cmax=1079[977,1192]90% m2, corresponding to a crown radius of 18.5 m. Although the best model did not include a covariate, the model including wood density provided a competitive fit (ΔWAIC =0.6, Table 1), with a slight positive relationship between the intercept and wood density (Fig. 3; r2=0.02 and 0.01 for the relationships presented). This suggests that species with high wood densities tend to have slightly broader crowns at all trunk sizes.

Individual tree aboveground biomass (AGB) estimates based on the community-average power model (H=3.02D0.56; Table S2) were strongly upwardly biased for large trees relative to estimates based on measured heights, whereas AGB estimates based on the gMM height model were unbiased (Fig. 4). Individual-level AGB estimates calculated using tree height predictions based on the power model exceeded those based on the gMM model by ever larger proportions at larger trunk diameters, with an overestimate of 10 % at D=66 cm [52, 80]90% that increases up to 59 % [51, 67]90% at D=250 cm (Fig. 4). This difference in individual-level AGB estimates for large trees translated into substantial differences for whole-plot AGB. Estimates of whole-plot AGB in trees with D≥ 1 cm using the community-average power model were 12.3 % larger than those using the better-supported community-average gMM model (283 vs. 252 Mg dry matter ha-1; Table 3). The incorporation of information about species identity reduces the difference between the models, with the power model estimate exceeding the gMM model estimate by only 4.5 % (276 vs. 264 Mg ha-1). As expected, deviations between estimates based on the power and the gMM models were more pronounced for larger diameter classes (Table 3).

Our analysis supported a saturating relationship between tree height and trunk diameter, consistent with theory (Falster and Westoby, 2003; Niklas, 2007) and with previous studies in tropical forests (Thomas, 1996; Bullock, 2000; Banin et al., 2012; Feldpausch et al., 2012; Molto et al., 2014; Fayolle et al., 2016; Ledo et al., 2016) and other forest biomes (e.g., Canham et al., 1994). The deceleration of height with respect to trunk diameter has been explained by multiple (mutually compatible) mechanisms including mechanical resistance (e.g., McMahon, 1973), growth and hydraulic constraints (e.g., Niklas and Spatz, 2004), and asymmetric competition for light (e.g., Iwasa et al., 1985; Bohlman and O'Brien, 2006; Falster and Westoby, 2003). Past work suggests that mechanical resistance to self or wind loading cannot explain tree height allometries, as trees are generally much shorter for a given diameter than the limits based on mechanical constraints (Niklas, 2007). Metabolic theories based on hydraulic constraints predict a constant logarithmic scaling between tree height and trunk diameter, with an exponent close to 2/3 (Niklas and Spatz, 2004; West et al., 2009), which is inconsistent with our results that show that the community-level power function exponents differ significantly from 2/3 and that the data diverge strongly from a power function.

Interspecific variation in tree height scaling parameters was associated with sapling growth rates, which suggests a tendency for shade tolerance and allometric strategies to be aligned in this community (Wright et al., 2010). At one extreme are the fast-growing, light-demanding tree species that are taller at small stem diameters; at the other extreme, slow-growing, shade-tolerant species are taller at larger diameters (Bohlman and O'Brien, 2006), with larger asymptotic heights (parameter a). This does not mean that shade-tolerant species tend to have larger maximum heights, because maximum heights depend on maximum diameters and are often much less than asymptotic heights for small-statured species (Fig. S1). The differences in allometric parameters should be interpreted in terms of differences in trajectories, especially at small diameters, where light-demanding species take greater risks by growing taller for a given diameter. In general, shade tolerance, and maximum height are largely independent axes of variation among tropical tree species (Bohlman and Pacala, 2012; Rüger et al., 2018) and may if anything tend to be negatively correlated across species (Poorter et al., 2006; Wright et al., 2010; Loubota Panzou et al., 2018). Our results quantify how variation in shade tolerance aligns with differences in height trajectories and thus in the parameters of saturating height allometric functions, thereby providing a basis for defining plant functional types representative of different gap-successional stages in tropical forests (Thomas, 1996; Falster et al., 2017).

Crown area and trunk diameter presented a scale-invariant relationship, with no indication of saturation even for the largest trees in our dataset. As a consequence, the model selection procedure favored power function models with estimates of the community-level exponent close to 4/3 (b=1.35[1.31,1.38]90%). This result is consistent with previous analyses across large scale environmental gradients reporting allometric exponents for crown area between 1.21 and 1.36 (Bohlman and O'Brien, 2006; Muller-Landau et al., 2006; Heineman et al., 2011; Antin et al., 2013; Blanchard et al., 2016). This large-scale consistency in community-level relationships emerges despite local variation among species (e.g., the exponent b ranged between 1.09 and 1.77 across species, Table S2). Modeling studies show that community-level crown area allometric parameters crucially determine the scaling of tree growth and mortality and the parameters of tree size distributions (Muller-Landau et al., 2006; Farrior et al., 2016). The fitted community-level crown area exponent is consistent with predictions of 4/3 scaling by elastic similarity models describing mechanical resistance to wind (McMahon, 1973) as well as by metabolic models invoking design constraints in transportation networks (West et al., 2009).

Our finding of high interspecific variation in the allometric scaling of crown geometry is consistent with previous studies (Iida et al., 2012; Lines et al., 2012; Blanchard et al., 2016). This interspecific variation has been linked to local differentiation and niche partitioning into canopy layers (Clark et al., 2008; Bohlman and Pacala, 2012). For instance, the crowns of subcanopy trees are wider than those of tall-statured trees on BCI (Bohlman and O'Brien, 2006). Our analysis of crown area favored an allometric model lacking trait influences on species-specific parameters, although the model featuring a weak positive relationship between the intercept of the power function and the average wood density of each species also received considerable support. The estimated relationship of crown area to wood density is consistent with the theory that high wood density enables more efficient horizontal crown expansion (Anten and Schieving, 2010) and with previous results for BCI (Francis et al., 2017) and Pasoh, Malaysia (Iida et al., 2012; Francis et al., 2017). However, overall wood density explains relatively little of the extensive interspecific variation in crown area allometries. Processes like crown plasticity in response to competition (Thomas, 1996; Poorter et al., 2008) and other traits may explain additional variation in crown geometry; for example, Loubota Panzou et al. (2018) found that wind-dispersed species had taller heights and larger crown dimensions.

Allometric models for individual trees remain the preferred method to estimate forest biomass and production at the stand level from plot data (Chave et al., 2014; Brienen et al., 2015) and provide a basic template to model carbon allocation, tree growth, and tree competition in dynamic vegetation models. Whereas many biomass models incorporate only individual trunk diameter (Brown, 1997) and species wood density (Brown et al., 1989), current state-of-the-art models typically include estimates of tree height as well (e.g., Chave et al., 2014). Crown dimensions have also been incorporated in some models (Goodman et al., 2014; Ploton et al., 2016), although Fayolle et al. (2018) found a minor role of either crown or height dimensions on biomass estimates. Inclusion of height and/or crown dimensions in tree biomass models reduces errors in biomass estimates, especially for large trees, which contribute disproportionately to forest biomass and function (Lindenmayer et al., 2012).

Our results confirm the importance for biomass estimation of accounting for saturation in height–diameter allometries (Feldpausch et al., 2011; Molto et al., 2014; Fayolle et al., 2016). If heights are not directly measured, any estimates of heights should be based on fitting saturating height functions to datasets with sufficient data for large trees to accurately capture the saturating component (Sullivan et al., 2018). The use of power function fits for heights leads to substantial overestimates of biomass of large trees, which translates to substantial overestimates of stand-level biomass.

The considerable heterogeneity among species in both tree height and crown area allometries presents another opportunity for improving estimates of tree biomass. The use of average allometric models that ignore changes in species composition can result in biased estimates of total biomass, reflecting the underlying nonlinearities of these relationships. At the same time, it is clearly impractical to develop species-specific allometries for every tropical tree species. The use of hierarchical models based on functional or demographic traits provides a manageable option for incorporating and accounting for the diversity of allometric scaling relationships in biomass models. Ideally, such hierarchical models would be grounded in a mechanistic understanding of underlying trade-offs on trait diversity (Falster et al., 2017).

Despite growing evidence highlighting the deceleration in diameter–tree height scaling (e.g., Thomas, 1996; Bullock, 2000; Banin et al., 2012; Ledo et al., 2016), the power function remains the most commonly used model of tree height allometry in tropical forests (e.g., Antin et al., 2013; Goodman et al., 2014; Blanchard et al., 2016; Mensah et al., 2018). Even studies featuring saturating relationships often fix the exponent of the gMM or Weibull functions to unity (e.g., Banin et al., 2012; Ledo et al., 2016; Fayolle et al., 2016; Molto et al., 2014), a value that our results show is inconsistent with data. Our results favored the gMM function over the modified three-parameter Weibull previously proposed by Thomas (1996), suggesting that it provides the required level of flexibility to accommodate changes in tree height scaling during ontogeny. Three-parameter saturating models clearly outperform two-parameter power functions in large datasets containing data on many large individuals; however, the advantage in fit of the saturating models is often insufficient to compensate for the penalty of an extra parameter in the many cases in which smaller datasets or those with data for few larger individuals are analyzed in isolation (Thomas, 1996; Iida et al., 2011; Goodman et al., 2014). We recommend that future analyses of small datasets on tropical tree allometries be conducted in a Bayesian framework in which prior data for larger datasets informs the choice of functional forms (i.e., restriction to saturating functions) and informs prior distributions on parameter values. Future studies can take advantage of our data, code, and results to constrain inferences on tree allometry using informative priors (e.g., Ellison, 2004).

Our analysis of crown allometric scaling involved an unusually large dataset, yet remained limited by sample size, measurement difficulties, and failure to address other dimensions of crown size such as crown depth. The crown area dataset was only one-fifth the size of our tree height dataset, and only 1/14 of the trees had trunk diameters greater than 100 cm. For ground-based data, which constituted the vast majority of our dataset, measurements of crown dimensions are more complicated and time-consuming than those of height, and we expect them to have higher measurement error. Aerial and even satellite imagery increasingly offers an alternative for precisely and accurately measuring crown areas of fully sun-exposed trees, an alternative we took advantage of here. However, these methods do not enable crown area estimates for subcanopy trees (but see e.g., Paris et al., 2016; Shendryk et al., 2016), which differ systematically in their crown allometries. Finally, we evaluated only crown area, even though crown depth and crown shape are also important for the estimation of tree biomass (Goodman et al., 2014; Ploton et al., 2016) and for characterizing tree species life history strategies (Canham et al., 1994; Bohlman and O'Brien, 2006; Poorter et al., 2006). Despite these limitations, our analysis consistently favored models of crown area vs. trunk diameter without saturation and suggested a weak effect of species differences in wood density on the considerable interspecific variability in the scaling of crown size.

Future allometric studies should address additional individual-level, species-level, and site-level covariates of tree allometry and develop improved models of their influences. Interspecific variation in allometry may be more fully explained by incorporating better measures of shade tolerance and additional traits such as seed dispersal mode, leaf habit (deciduous or evergreen) and maximum stature (e.g., Poorter et al., 2003, 2006; Loubota Panzou et al., 2018). Interindividual variation in allometry depends not only on species identity but also on environmental conditions, biogeographic region, and competitive neighborhood. Previous studies show that tropical tree height allometries vary with climate (Chave et al., 2014; Mensah et al., 2018), topography (Ferry et al., 2010; Marshall et al., 2012), edaphic conditions (Aiba and Kitayama, 1999; Feldpausch et al., 2011), canopy position (Thomas, 1996; O'Brien et al., 1995; Poorter et al., 2006), and light exposure (Rüger et al., 2012). Ideally, these factors would be incorporated not in a purely phenomenological manner but would be informed by mechanistic models of underlying trade-offs and alternative strategies (Dybzinski et al., 2011; Farrior et al., 2013).

Tree allometric functions are critical components of forest biomass estimates and of mechanistic models of forest structure and dynamics. Forest size structure and biomass in vegetation models are highly sensitive to allometry parameters and functional forms (Farrior et al., 2016; Weng et al., 2017). Our results confirm that allometric models for tropical trees should incorporate saturating functions for tree height and interspecific variation in scaling parameters. In our analyses of data for over 10 000 tropical trees, tree height presented a saturating relationship with trunk diameter that was well captured by the three-parameter generalized Michaelis–Menten or Weibull functions, whereas power function models exhibited systematic biases in tree height predictions, especially for large trees. In contrast, our somewhat smaller dataset for crown area exhibited a constant scaling with stem size, consistent with a power function. We observed extensive interspecific variability in allometric scaling, with shade tolerance explaining considerable variation in height parameters and wood density weakly related to crown area parameters. The relationship of tree allometric parameters with functional and demographic traits can provide a basis for the incorporation of compositional effects into estimates of forest biomass and production, including through the improved parameterization of tropical plant functional types in vegetation models.