The Influence of Ecosystem and Phylogeny on Tropical Tree Crown Size and Shape

The sizes and shapes of tree crowns are of fundamental importance in ecology, yet understanding the forces that determine them remains elusive. A cardinal question facing ecologists is the degree to which general and non-specific versus ecological and context-dependent processes are responsible for shaping tree crowns. Here, we test this question for the first time across diverse tropical ecosystems. Using trees from 20 plots varying in elevation, precipitation, and ecosystem type (savanna-forest transitions) across the paleo- and neo-tropics, we test the relationship between crown dimensions and tree size. By analyzing these scaling relationships across environmental gradients, biogeographic regions, and phylogenetic distance, we extend Metabolic Scaling Theory (MST) predictions to include how local selective pressures shape variation in crown dimensions. Across all sites, we find strong agreement between mean trends and MST predictions for the scaling of crown size and shape, but large variation around the mean. While MST explained approximately half of the observed variation in tree crown dimensions, we find that local, ecosystem, and phylogenetic predictors account for the half of the residual variation. Crown scaling does not change significantly across regions, but does change across ecosystem types, where savanna tree crowns grow more quickly with tree size than forest tree crowns. Crowns of legumes were wider and larger than those of other taxa. Thus, while MST can accurately describe the central tendency of tree crown size, local ecological conditions and evolutionary history appear to modify the scaling of crown shape. Importantly, our extension of MST incorporating these differences accounts for the mechanisms driving variation in the scaling of crown dimensions across the tropics. These results are critical when scaling the function of individual trees to larger spatial scales or incorporating the size and shape of tree crowns in global biogeochemical models.

Ecologically, tree crown size and shape may be subject to multiple tradeoffs, including lateral 66 extension for light interception and competitor suppression versus mechanical risk, leaf exposure 67 versus drought risk, and fast versus slow growth strategies (Verbeeck et al., 2019). The above 68 general theories do not deny mechanical and other constraints. Instead, adaptive differentiation 69 either does not figure importantly in them or plays a secondary role. Nonetheless, general theory 70 provides a baseline by which to assess their underlying assumptions and hypothesized drivers of 71 variation in canopy dimensions. 72 Ecological theory appears best able to explain sapling crown form, when variation in crown 73 shape and form is determined more by genetics than stochastic and local processes (e.g. King,74 1998; Sterck and Bongers, 1998). Understanding variation in adult crown form, however, has 75 proven more difficult for a number of reasons. These reasons likely include the fact that 76 numerous architectural paths can lead to similar forms (e.g. Fisher and Hibbs, 1982); the 77 measurement of adult crowns is more difficult than that of juvenile crowns; and general and 78 stochastic processes may overpower competitive and genetic ones, making the latter effects 79 difficult to detect. can exceed an order of magnitude (Poorter et al., 2015). We consider how the inclusion of 85 information on ecosystem context, biogeographic region, and phylogenetic position can help 86 explain this large variance. 87 Beyond their fundamental importance in understanding plants and forests, tree crown shapes and 88 sizes are of increasing interest to communities developing dynamic global vegetation models 89 (DGVMs). Representations of canopy structure in these models range from 1-D "single leaf" 90 models, to 2-D multilayer models (e.g. perfect plasticity approximation Purves et al., 2007), to 91 efforts to fuse individual-based models with DGVMs to represent the influence of variable 92 crown allometry on ecosystem dynamics (Fischer et al., 2019). Crown allometries play a role in 93 how trees assemble and compete in forests, so as DGVMs become more complex and attempt to 94 represent individual dynamics, models for crown allometry and plasticity of those allometries are 95 needed. 96 In this study we examine general patterns in tree crown size (width, depth, surface area, and 97 volume), shape (relative depth), and allometry across an original dataset of 1144 individual tree 98 crowns of 281 species spanning one elevation transect, one savanna-forest-precipitation gradient, 99 and one savanna-forest gradient with constant precipitation, across the paleo-and neo-tropics. 100 We evaluate predictions and underlying assumptions of MST theories in light of our data, and 101 then examine the variance around MST fits from an ecological vantage point. To evaluate the 102 relative contributions of general and context-specific processes, we ask if ecological predictors 103 can improve general models. We pose ecological hypotheses across ecosystems, space, 104 biogeography, and evolutionary history, and test them in conjunction with general predictors 105 ( Figure 1, Table 1.). To our knowledge, this study represents the first test across ecosystems, 106 and the first test across environmental gradients, of crown size and shape in the tropics. (1) 112 Second, West et al. (2009) assume that crown radius scales isometrically with tree height: 113 Third, by substitution, the relationship between crown and stem radius is derived as: 114 We refer to the first two assumptions (Eqs 1, 2) as distal assumptions, and the derived 116 predication of (Eq 3) as the proximate prediction. By further assuming a spherical, 117 Euclidean-uniform (as opposed to fractal; Voss, 1988 Next, using the residuals of our scaling models, we examine how the variance from the mean fit 130 is partitioned across taxonomic (genus and family), ecosystem type, spatial (plot), and 131 biogeographic (regional) groups. We then examine those groups in more detail. Our study employs 20 one-hectare plots across the three environmental gradient transects above 168 (Table S 1 y -1 to 5302 mm y -1 across all sites along the gradient ( census and diameter data, a sampling protocol was adopted wherein species were sampled that 210 maximally contributed to plot basal area (a proxy for plot biomass or crown area). We aimed to 211 sample the minimum number of species that contributed to 80% of basal area, although in the 212 diverse lowland forest plots we only sampled species comprising 50-70% of plot basal area. 213 Within each species, 3-5 individual trees were chosen for sampling (5 trees in upland Peruvian 214 sites and 3 trees in lowland Peruvian, Brazilian, and Ghanaian sites; Table S 2). If 3 trees were 215 not available in the chosen plot, we sampled additional individuals of the same species from an 216 area immediately surrounding the plot. 217 Tree climbers sampled a fully sunlit canopy branch at least 1 cm diameter of each tree, from 218 which simple leaves, or individual leaflets from compound-leaved species (both referred to as 219 'leaf' below), were removed and measured. In the case of compound leaves, the entire 220 compound leaf was also collected for whole-leaf area calculations. Branches and leaves were 221 chosen with minimal damage (e.g. from herbivory). Five leaves per branch were sampled in 222 Peru, and three per branch were sampled in Brazil and Ghana. 223

CROWN MEASUREMENT 224
Tree crowns were measured using a laser hypsometer (TruPulse 360/360R, Laser Technology  225 Inc., Colorado, USA). In "Missing Line" (ML) mode, the TruPulse 360 returns horizontal 226 distance (HD), vertical distance (VD), and azimuth (AZ) between two points in space as 227 determined by 2 laser pulse returns that determine distance coupled with azimuth measurements 228 from the unit's internal compass (TruPulse 360/360B User's Manual, Laser Technology Inc., 229 Colorado, USA). We took ML measurements between stem base and crown top, crown bottom, 230 and usually 6-20 points around the circumference of the crown perimeter depending on the 231 complexity of its shape. Because we were interested in tree function (e.g. gas exchange 232 capacity), we defined the crown base as the lowest significant foliage that was not a resprout or 233 otherwise relatively spurious, instead of using the first primary branch. We applied a convex 234 hull to this set of points to yield a 3-D polyhedron from which crown dimensions were extracted. 235 We estimate crown volume and surface area as the volume and surface area of the convex hull. 236 We estimate average crown radius as the radius of a circle with the same area as the 2-D convex 237 hull of the points when projected onto a horizontal plane. 238 Manual measurements of crown widths and tree heights were taken using clinometer and tape 239 measure in a subset of Ghanaian trees to verify that results from the laser hypsometer method are 240 comparable to those from more widely used techniques. Dimensions from both methods 241 corresponded closely (N = 20; width R 2 adj = 0.83, SE = 1.06; height R 2 adj = 0.74, SE = 2.77), and 242 are within the range for the tangent method reported by Larjavaara and Muller-Landau (2013). 243 Larjavaara and Muller-Landau (2013) recommend the laser hypsometer method as opposed to 244 manual clinometers for tree height measurement, especially for cross-site studies where 245 instrument operators are different. They further note that while errors are inherent in both 246 methods, those introduced by the laser hypsometer may lend greater weight to leaf versus fine 247 wood structure, which largely corresponds with our goals here. assume that leaf area densities (LADs) of tree crowns are equivalent across species and tree size. 254 If these assumptions do not hold across species then we must control for this variation in one of 255 two ways: either measure the key characteristics in each individual or species, or evaluate scaling 256 parameters within (intraspecific), not across (interspecific) species. In effect, without 257 independent measurements of these characteristics, MST's assumptions compel us to model 258 scaling intraspecifically. In this study, we implement intraspecific LMMs by including species 259 as a random effect. Random effects enable us to account for variation between species with 260 sample sizes of 3-5 individuals per species. In our standardized major axis (SMA) regressions 261 (see Appendices), we group by species, and then examine the estimated overall fit. 262

NOTATION 263
Our notation of model coefficients and parameters uses α when referring to exponential (or 264 scaling) parameters, and when referring to linear (or normalization) parameters. Subscripts 265 may include a comma, such as α , , in which case the first term (y) indicates the dependent 266 variable in the model, and the second term (x) indicates the independent variable that coefficient 267 is associated with. Thus, α ℎ,ℎ indicates the exponential parameter of tree height when 268 predicting crown depth. Since rstem is the most common predictor, we omit commas for 269 parameters associated with rstem. 270

STATISTICAL AND PHYLOGENETIC MODELS 271
We employed both LMM and SMA regressions to estimate scaling parameters and test 272 hypotheses. Our primary analyses are conducted with LMM, and confirmed with SMA in the 273 Appendices. The LMM approach is advantageous when accounting for variation across many 274 groups (e.g. species) via random effects, and it is appropriate for fitting allometry models 275 (Kilmer and Rodríguez, 2016 LMM log-log models suffice for scaling tests with single predictors, but when multiple 296 independent non-linear predictors are necessary, one must either accept a multiplicative 297 relationship between predictors, or resort to non-linear mixed models (NLMM). We use 298 NLMMs below when testing multiple non-linear predictors that cannot be expressed as a 299 modification to an existing coefficient (see SI). When using NLMMs below, we utilize the nlme 300 R package (Pinheiro et al., 2016), and we include species-level random effects for each predictor 301 unless indicated otherwise. When NLMMs would not converge, we resorted to LMMs. 302 MST predicts a lack of relationship between tree size and relative crown depth. We therefore fit 303 both linear and log-log models when evaluating relative depth (

ALLOMETRIC SCALING OF TREE CROWNS 331
Here we fit MST models to our data and compare our empirical scaling exponents with 332 theoretical predictions (H1 -H3). Model equations are specified in Table 2. SMA results largely  333 confirm LMM results, and details are included in the appendices. 334

ASSESSING METABOLIC SCALING THEORY'S DISTAL ASSUMPTIONS 335
The two assumptions underlying MST crown scaling, namely 2/3 scaling between stem radius 336 and height and isometric scaling between height and crown radius, are both violated in our 337 dataset. Instead, the constant stress model ( ℎ = 1/2) better explains the relationship between 338 stem radius and height (Model 2; ℎ = 0.46, 95% CI 0.42 -0.51; Figure S  The assumption of height-crown isometric scaling (Eq. 2), in particular, lacks empirical 343 underpinnings. Instead, we propose an alternative assumption to better account for the drivers of 344 tree form and function. Specifically, we allow crown radius to depend on both and ℎ 345 independently by modifying the second assumption (Eq. 2) to include (Model 4; see 346 Appendices for derivation). Thus, the modified assumption becomes: 347 see Models for crown depth in Appendices for details). We therefore use the model with both 365 predictors, but report the results from themodel with just stem radius for copmrison with the 366 scaling models for other crown dimensions. 367 Crown depth scales with stem radius as α ℎ = 0.50 (Model 10, Figure S 21, Table 3) and as 368 α ℎ = 0.18, α ℎ,ℎ = 0.67 when also including tree height as a predictor (Model 12, Table  369 3). Remarkably, in this formulation, crown depth scales exactly with tree height as crown width 370 scales with stem radius. We do not support MST's prediction that crown depth scales with stem 371 radius as 2/3. Rather, our data suggest that crown depth scales with tree height as 2/3. 372

EXPLAINING RESIDUAL VARIATION IN SCALING 373
Taking the residual errors of the MST fits ( ), we examine how this variation is structured 374 across taxonomy, space (plots), biogeography (region), and ecosystem type (Figure 3, Table S 7). 375 We find that residual variation in crown width leaves ~50% unexplained, while biogeography 376 accounts for 28%, taxonomic family for 12%, and spatial (between plot) variation for 11%. 377 Crown volume and surface area residuals are similarly structured: unexplained variation 378 accounts for about half of the total, spatial variation account for ~20%, ecosystems for ~15%, 379 biogeography for ~10%, and taxonomic family for 7%. Crown depth residuals (Model 12) were 380 structured across space (24%) and biogeographic region (17%), with no taxonomic or ecosystem 381 signal.. 382 trees in the Chrysobalanaceae and Primulaceae, and some Moraceae taxa, tend to be smaller than 392 expected. These patterns do not seem to be structured by biogeography. That is to say, clades 393 with significantly larger or smaller crowns are largely comprised of species from all three 394 regions. 395

TRENDS ACROSS ECOSYSTEMS AND BIOGEOGRAPHIC REGIONS 396
Here we examine patterns and test hypotheses of crown scaling patterns across environmental 397 gradients (H6 -H8) and biogeography (H5). We discuss our observations of these trends in 398 detail in the Appendices (see Observed Trends across ecosystems and biogeographic regions). scaling. Linear models were generally unimproved by the addition of the biogeography predictor 408 (Table S 8). The variance partitioning models, however, attribute substantial variance to 409 biogeography, and especially in the case of crown width scaling (Figure 3). The unusual 410 formulation necessitated by log-log models with covariates (see Appendices) may explain this 411 disagreement. We conclude that while biogeography is associated with some variance, it is a 412 weaker influence than ecosystem type. 413

ENVIRONMENTAL GRADIENT HYPOTHESES 414
The observed trends across environmental gradients are described in detail in the Appendices 415 (see Observed trends across gradients within regions). Here we evaluate our specific hypotheses 416 related to the environmental gradients our study spanned. 417

Savanna-Forest Transitions 418
Savanna trees are shorter than forest trees for the same girth (Figure S 33). Consequently, 419 because crown depth scales principally with tree height, a 30cm DBH savanna tree is shorter and 420 has a shallower crown than a 30cm DBH forest tree (Figure S 32). When controlling for tree 421 height however, a 15m-tall savanna tree crown is more than 50% deeper than a 15m-tall forest 422 tree (Figure 7). Stout savanna trees with deep crowns are consistent with our hypothesis H6 423 (Open growth form). 424

Precipitation Gradient 425
We tested for effects of precipiation on crown depth across the Ghanaian transect, and across our 426 entire study. No precipitation covariates were significant across the Ghanaian transect (Table S  427 9, Table S 10). Modeling the effect of precipiation on crown depth across all sites, just one 428 precitation covariate was marginally significant ( �α ,ℎ � = 0.07; Table S 9, Table S 10).  429 Taken together, our analyses do not support hypothesis H7 (Depth drought tolerance). 430

Elevation Gradient 431
We hypothesized that crowns will become deeper as one moves from the low productivity sites 432 in the Andes down to the high productivity sites in the Amazon (H8 (Ecosystem speed)). Crown 433 widths decreased going downslope for small (10cm) and mid-sized (30cm) trees ( Figure   and that this tendency is influenced by ecosystem and evolutionary context. Scaling did not 444 change significantly across regions, but it did across ecosystem types with savanna crowns 445 growing more quickly with tree size than those of forests. We found that crown radius scales 446 with stem radius (King, 1996) whereas crown depth scales with tree height, and because tree 447 height differs more across ecosystems than stem girth, crown depth varies more across gradients 448 that crown depth. Controlling for tree size, crowns of legumes were wider and larger than those 449 of other taxa. Implications of specific results are discussed below. 450 MST predictions rely on chains of relationships. In this case, the linkage is between stem radius, 478 stem height, and ultimately crown width. We modified MST's stem height/crown width 479 assumption to include stem radius (Model 4), and found that the modification resolved the 480 previously-incongruent chain of scaling exponents (see Appendices, Resolving the incongruence 481 of MST distal assumption deviation and MST proximate prediction accuracy). We propose that 482 the following assumptions be used instead of equations 1 and 2: 483

METABOLIC SCALING THEORY
where ℎ = 1/2. Empirically, we find that ,ℎ = 0.16 and ′ = 0.59, but these values 486 should be informed by further development of theory before offering them as MST assumptions. 487

PHYLOGENETIC VARIATION 488
While crown scaling did not vary amongst most families, we do find a strong, consistent, and 489 biogeographically-wide phylogenetic signal in tree crown allometry in the large Fabaceae 490 crowns, and in particular, in the Mimosoideae and Papilionoideae subfamilies. What is it about 491 these subfamilies that could lead to their particularly wide crowns? 492 Of the three legume subfamilies, rhizobial associations that enable nitrogen fixation in root 493 nodules are common in the Mimosoideae and Papilionoideae, but less so in the Caesalpinioideae 494 (Allen and Allen, 1981;Andrews and Andrews, 2017). Given the correspondence between 495 increased crown size and rhizobia occurrence, the simplest hypothesis is that the increased 496 availability of photosynthate in N-fixers enables more carbon allocation to crown mass and 497 hence crown width. Alternatively, even if rhizobia do not affect overall per-tree productivity, 498 they might allow N-fixers to differentially allocate more carbon to crowns than roots. Finally, 499 some evidence suggests that root symbionts may stimulate increased leaf-level photosynthesis 500 through their role as carbon sinks (Kaschuk et al., 2009). Could such a sink-effect be reflected 501 on a whole-plant scale? If crown sizes are matched to the metabolic requirements of the 502 organism, and if these requirements grow as a result of root symbioses, then we might expect this 503 pattern of larger crowns in species that have root symbioses. This question would be an 504 interesting avenue for future research. 505 Large crowns may be advantageous by increasing carbon uptake due to increased leaf area and 506 light interception, but also by suppressing neighboring crowns. Taylor et al. (2017) found that 507 N-fixing trees inhibited growth of their neighbors and of plots where they were abundant. By 508 harboring large crowns, these N-fixers may be shading out their neighbors. 509 Does crown scaling differ between species? Iida et al. (2011) found that crown scaling 510 parameters did not differ more than expected from the community tendency in Pasoh, Malaysia. 511 Here we find that scaling in different forest types does differ (e.g., Figure 2.d), and that species 512 differ as well (e.g. Figure S 18). Indeed, there was a clear phylogenetic signal in allometric 513 scaling parameters (Figure 4). 514 While foresters have been well aware of species-specific crown shape since the inception of their 515 practice (Larson, 1963), the relationship between phylogeny and crown shape has remained 516 largely unexplored until now. Incorporating more studies across an even wider range of 517 ecosystems and environments into larger phylogenies would be likely to yield interesting insights 518 into the evolutionary pressures on the shapes of tree crowns. 519

ECOSYSTEM, BIOGEOGRAPHY, AND PHYLOGENY 520
Our community-wide hypothesis that crown depth decreases with elevation (increasing moving 521 downslope) in our Peruvian transect was based on Horn's (1971)  second mechanism, tree height, is indeed closely coupled with both relative and absolute crown 535 depth. It seems clear that the driver of increasing crown depths moving downslope is the 536 lengthening of height allometries from the uplands to the lowlands. This proximate "allometric" 537 driver does not exclude the first mechanism, ecosystem productivity, as a more distal driver. 538 Ecosystem productivity also shows a similar pattern to crown depth, and may be linked tree 539 height allometries. Indeed, NPP and mean vegetation height exhibit similar patterns across the 540 Peruvian transect (Malhi et al., 2017; Table S 1). 541 Crown depth exhibited a consistent response to ecosystem type. This suggests that crown depth 542 is adaptive, but given that H7 (drought-depth hypothesis) was not supported, precipitation is not 543 the primary factor driving it. We examined height -DBH allometries and crown depth 544 comparisons for growth form differences, and found support for hypothesis H6 (Open growth 545 form). Thus, savanna trees exhibit open-growth forms: they are ultimately shorter and harbor 546 shallower crowns than forest trees of equivalent girth, but have deeper crowns than forest trees of 547 equivalent height. 548 Biogeography has little effect on crown shape according to our analyses, in contrast to Moncrieff 549 et al. (2014). Thus, we should expect neither deep phylogenetic patterns in nor strong influence 550 of disparate faunal communities on crown shape. Rather, crown shape is likely more structured 551 ecologically and competitively, and if evolution does play a role, we might expect crown 552 architecture to be relatively plastic evolutionarily. 553

CONCLUSIONS 554
In this first look at tropical tree crown allometry across ecosystems, biogeography, and 555 phylogeny, we find that both the lack of patterns in some instances (radius predictions across 556 ecosystems and biogeography) and strong patterns in others (depth predictions across 557 ecosystems, leguminous crown size) spurs further questions. In particular, ecological patterns 558 such as ontogenetic variation and competitive effects on crown size may explain some of the 559 observed patterns. Plasticity of crown dimensions in relation to local competition will be 560 important to quantify for models to effectively simulate local dynamics, and this should comprise 561 future research as well. 562 Our study lacks data from the Asian tropics. Thus, while we did not find a strong biogeographic 563 signal in crown allometry, future studies that do include them and their especially tall trees may 564 find one. 565     Figure S 19. for SMA fits vs data). Model includes random intercepts and slopes per species. Error bars indicate 95% confidence intervals fit by likelihood surface profiling. Categorical site variables were encoded as deviation contrasts, which fits the magnitude of the site coefficients as the mean of the slope at that site minus the grand mean slope across all individuals. In order to aid interpretation, we added the grand mean to each site effect and its confidence interval. The Stem Radius effect Drivers of Tropical Tree Crown Size represents the main effect of stem radius on crown radius, and in this context is the grand mean across all individuals, taking the species random effects into account. Figure 3. Variance associated with species' deviation from mean crown scaling relationship The quantity analyzed here is the mean species residual, or deviation, from the _ = scaling model. LMM models were re-fit with site as a random effect, and residuals were obtained with the random effects removed. Figure 4. Mean per-species residuals from the crown volume vs stem radius scaling LMM with species as a random effect mapped onto a tree species phylogeny (see Methods for details of phylogeny construction). A residual in this instance implies a difference in intercept, not slope, in the model. Size of circle corresponds to size of residual. Internal node states were determined using fastANC (see Methods), with colors indicating direction and confidence of internal node estimates: 95% confidence interval of model residuals of grey nodes intersects zero, green nodes indicate clades with larger than expected crowns, and blue nodes clades with smaller than expected crowns. Tips are not evaluated for significance and are therefore not colored. Similar plots with species names are included in SI ( Figures S 22 -S 25).  . Modeled crown dimensions and 95% CIs for a 30cm DBH tree. The crown depth model also included tree height, which was fixed in (d) using our height vs stem radius allometry (see SI) and averaged across sites so the same DBH and height were used for each site, and allow to vary in (e) and (f) based on the mean per-site height -DBH allometry (Model 2). Predictions are derived from the following LMM models described above: