The structure of tropical forests and sphere packingsFranziska Tauberta,1, Markus Wilhelm Jahnb, Hans-Jürgen Dobnerc, Thorsten Wieganda,d, and Andreas Hutha,d,e

aDepartment of Ecological Modelling, Helmholtz Centre for Environmental Research - UFZ, 04318 Leipzig, Germany; bGeodetic Institute, Department of CivilEngineering, Geo- and Environmental Sciences, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany; cFaculty of Computer Science, Mathematicsand Natural Sciences, University of Applied Science–Hochschule für Technik, Wirtschaft und Kultur (HTWK), 04277 Leipzig, Germany; dGerman Centre forIntegrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, 04103 Leipzig, Germany; and eInstitute for Environmental Systems Research, Department ofMathematics/Computer Science, University of Osnabrück, 49076 Osnabrück, Germany

Edited by Robert D. Holt, University of Florida, Gainesville, FL, and accepted by the Editorial Board October 20, 2015 (received for review July 8, 2015)

The search for simple principles underlying the complex architectureof ecological communities such as forests still challenges ecologicaltheorists. We use tree diameter distributions—fundamental for de-riving other forest attributes—to describe the structure of tropicalforests. Here we argue that tree diameter distributions of naturaltropical forests can be explained by stochastic packing of treecrowns representing a forest crown packing system: a method usu-ally used in physics or chemistry. We demonstrate that tree diame-ter distributions emerge accurately from a surprisingly simple set ofprinciples that include site-specific tree allometries, random place-ment of trees, competition for space, and mortality. The simplestatic model also successfully predicted the canopy structure, re-vealing that most trees in our two studied forests grow up to 30–50 m in height and that the highest packing density of about 60%is reached between the 25- and 40-m height layer. Our approach isan important step toward identifying a minimal set of processesresponsible for generating the spatial structure of tropical forests.

tropical forest | forest size structure | stochastic geometry |tree crown packing | leaf area

Forests are one of the world’s best investigated ecosystems(1–4). However, despite all of the studies devoted to forests,

mechanistic connections among important features of forestphysiognomy are not fully understood. Of crucial importance forthis are tree diameter distributions, which have been used fordecades in ecology and forestry to characterize the state of for-ests. Tree diameter distributions are available for many forests ofthe world and allow together with tree allometries prediction ofother important forest attributes like leaf area (5), basal area (6),above-ground biomass (7), tree density, and the presence orabsence of disturbances (8). Tropical forests usually include manysmall trees and far fewer large ones (1, 9). Diameter distributionscan also be predicted from dynamic forest models (10–13) in whichthe forest structure emerges from the interplay between the dynamicprocesses of mortality, regeneration, competition, and growth.In this study, we argue that tree diameter distributions of natural

tropical forests can be predicted by stochastic packing theory—amethod usually used in physics or chemistry—together with site-specific tree allometries. Packing systems have a long history ofuse across a wide range of disciplines, going back as far asKepler’s analysis of regular packing of spheres (i.e., with amaximum of 74% filled volume). Irregular stochastic packing ofspheres has been analyzed in the last decades and shows lowermaximal packing densities (e.g., jammed sphere packing systemsin physics, with 55–64% filled volume) (14, 15). We show herethat simple principles of stochastic packing theory together withtree allometries and other simple structural rules allow for anaccurate prediction of observed tree diameter distributions andrelated structural attributes for two tropical forests in BarroColorado Island (BCI, Panama) and Sinharaja (Sri Lanka). Bothforests show major differences in climate, soil, and topographythat affect forest structure and dynamics (16, 17).Our approach consists of three parts. First, we use tree al-

lometries to transform size data of a tree (including the positionx and the stem diameter d, which are available from forest

inventories or from the packing model) into a simplified treewith a crown modeled as sphere with radius c(d) and height h(d)(Fig. 1). We use the simple allometric relationships c(d) = c0 d

pc

and h(d) = h0 dph with parameters c0, h0, ph, and pc determinedfrom independent datasets (18, 19) (for details, see Methods).Other height-diameter relationships saturating in maximum treeheight (7) could be used alternatively. However, because only fewtrees become large enough to experience this saturation, this haslittle influence on our results. The approximation of crowns asspheres is a simplification, given that crowns might shape differ-ently during growth (20) (Fig. S1). In SI Results, we examined theinfluence of the assumed crown shape on our results (Fig. S2C).Second, we simulate the static forest packing system (FPM)

based on the following rules:

i) A tree with a stem diameter d is randomly determined froma uniform distribution within the interval [dmin, dmax] (withdmin = 1 cm and dmax corresponding to the maximal treeheight). Fig. S3 shows that our results are insensitive to theselection of dmin and dmax.

ii) We accept a tree of stem diameter d only with probabilityp(d) (i.e., the survival probability of a tree growing fromstem diameter dmin to diameter d). This rule thus considersaverage tree mortality and growth. For simplicity, we as-sumed constant annual stem diameter growth rates g andconstant annual mortality rates m that yield a probabilityp(d) = p0 exp[−(m/g)d] (Methods). For comparison, we in-vestigated in this study two versions of our model: one with-out mortality (i.e., m = 0) and one with mortality.

iii) Based on the allometric relationships between tree heighth(d) and crown radius c(d), the stem diameter d is translatedinto a crown with center h(d) − c(d) above ground (Fig. 1 Aand B). Thus, in contrast to irregular stochastic packing, eachcrown is located in a given height layer (depending on d). Thisheight constraint will reduce the packing density.

iv) We tentatively locate the stem and crown (and thus the tree)at a random position within the study area.

Significance

Explaining the tree size structure of tropical forests is of crucial im-portance because other important forest attributes can be derivedfrom this. The tree diameter distribution, for example, determinesthe amount of carbon stored in a forest. Here we present a simpleand powerful approach based on stochastic geometry and tree al-lometries that can be used to predict tree diameter distributions.

Author contributions: F.T. and A.H. designed research; F.T. and M.W.J. performed research;F.T., M.W.J., H.-J.D., T.W., and A.H. contributed new reagents/analytic tools; F.T. analyzeddata; and F.T., M.W.J., H.-J.D., T.W., and A.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. R.D.H. is a guest editor invited by the EditorialBoard.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: franziska.taubert@ufz.de.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1513417112/-/DCSupplemental.

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v) The tree is retained if it does not overlap with an alreadyplaced tree (i.e., their crowns and stems).

vi) The algorithm stops if the observed number of individualtrees is placed.

Thus, the spatially explicit crown packing system emerges frommodel rules and simple site-specific tree allometries that are pa-rameterized from the data or the literature (for details, see Meth-ods). Results of a sensitivity analysis are provided in SI Results.Finally, in the third step, the crown packing systems are analyzed

with methods of classical packing theory in terms of the space oc-cupied by crowns (Fig. 2A), local forest height (Fig. 2B), and thedistribution of leaf area and packing density at different heights andspatial scales (Fig. 2 C–E). To calculate the leaf area of crowns, weassumed for simplicity that they are homogeneously filled with leaves(see SI Results for results if we modify this assumption; Fig. S2D).

ResultsOn average only 15% of the available space is occupied by treecrowns at BCI (Fig. 2A) and 20% at Sinharaja (Fig. S4E; analysisof packing density φ at the local scale of 0.04 ha for 1,250 sub-plots at BCI and 625 subplots in Sinharaja). Nevertheless, insome plots, we observe higher densities, and in other plots, lowerdensities (e.g., due to former gaps of large fallen trees). Forexample, in the BCI forest, the crown packing density at the localscale ranges from 3% to 53% (Fig. 2A). The local packing den-sities are much smaller than the optimized regular packing ofmonosized spheres and below its irregular counterparts, which areknown from stochastic geometry theory (see below) (14). Clearly,this is because tree crowns can only be located in their associatedheight layer and because we assumed a maximum tree height of60 m, which is rarely attained (Figs. 1 and 2B). A reduced maximumtree height of 45 m would increase the overall packing density forBCI only from 15% to 18%. Ecologically more meaningful mea-sures are therefore the crown packing density across differentheight layers (Fig. 2C) and the associated distribution of leaf

area density (Fig. 2D). Essentially, we find a unimodal localcrown packing distribution (Fig. 2C).The densest layer in the BCI forest is reached at a height of

about 25 m (40% of forest height) and yields a local mean leaf areadensity of lmax = 0.6 m2/m3 (Fig. 2D, Fig. S5 G and H, and TableS1). Interestingly, the densest height layer is close to maximalstochastic packing densities (i.e., 60%; jammed packing), for whichthe filled space ranges from random loose (55%) to random close(64%) sphere packings (14). Note that lmax and the correspondingheight also vary locally (Fig. S5 G and H). We obtained similarresults for the Sinharaja forest (Figs. S4 G and H and S6B andTable S1). The packing density and leaf area distributions wereunimodal at the Sinharaja forest (Fig. 3E), but that of BCI showeda small additional peak at about 17% (∼11 m) of forest height (Fig.2D and Fig. S5H). We suspect that this is caused either by differ-ences in soil quality or by former treefall gaps at the BCI forest. Ona larger scale, the leaf area distribution is increasingly influenced bytopography. We suspect that the small additional peaks in Fig. 2Eare caused by the specific tree allometries for the BCI forest pro-ducing a relatively high number of mid- to large-size trees, whichthen allow only much smaller trees to be placed below (Fig. 1).The forest packing model without mortality (i.e., m = 0) sub-

stantially overestimated the leaf area density at larger heights andthe number of large trees (Fig. 3 A and B and Table S2) and cantherefore be ruled out. In Sinharaja, this overestimation also holdsbut to a much lesser extent (Fig. S7 and Table S2). However, themodel using the observed mean mortality and growth rates suc-cessfully reproduced the observed size structures of the tropicalforest in BCI (Fig. 3D), the leaf area distribution (Fig. 3C), andthe local scale variation of the packing density (0.04 ha; Fig. 4A

Fig. 1. Representations of a tropical forest on BCI as a sphere packing system.Comparison of the sphere packing systems predicted by the model (A and B) andderived from the field data (C and D). We show 1-ha plots (A and C) and corre-sponding 100 ×10-m plot-transects (B andD) through the simulated and observedBCI forest. The higher canopy is only sparsely filled with tree crowns (B and D).

Fig. 2. Analysis of structural attributes for the BCI forest (based on in-ventory data). (A) Frequency distribution of the local crown packing densityφ (in %) where φ is the proportion of space occupied by tree crowns (analysisof 20 × 20-m subplots, n = 1,250; using 2% class widths). (B) Frequencydistribution of local forest height (in % of assumed maximum tree height)estimated for 20 × 20-m subplots (using 4% class widths). Green solid verticallines represent mean values. (C) The average local packing density (dots) andSD (gray horizontal lines), estimated for different height layers (20 × 20-msubplots, using 1-m layer widths). (D) Same as C, but for the estimated leafarea distribution. (E) Leaf area distribution at different heights (using 1-mlayer widths) estimated for the entire 50-ha forest.

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and Table S1). We found at BCI slight differences in maximumleaf area density (Fig. S5G) and in the densest height layer atwhich this maximum is reached (Fig. S5H). This difference couldbe caused by canopy gaps in the data due to fallen larger trees,which are not considered in the model. The results for the Sin-haraja forest are similar (Figs. 3 E and F and 4B, Fig. S4 G and H,and Table S1), but here our model slightly underestimates leaf areadensity at heights below 20% of forest height (Fig. 3E), and theobserved local packing density at 20 × 20-m subplots is somewhatwider than the model prediction (Fig. 4B). We suspect that themismatch for the Sinharaja data (especially for the local packingdensity; Fig. 4B) was caused by our assumed tree allometries, whichare only approximations. Finally, the posterior tree diameter dis-tributions emerging from our simulation (Fig. 3D and F) are largelydifferent from the prior p(d) used to propose tentative trees inthe sphere packing algorithm (rule 2). This difference outlinesthe importance of the additional rule 5 of stochastic geometry toselectively accept proposed trees depending on their size.

DiscussionOur tree crown packing simulations showed that simple principlesof stochastic geometry combined with site-specific tree allometriesand tree mortality are sufficient for accurately predicting impor-tant attributes of tropical forests such as tree diameter distributionand leaf area distribution (Fig. 3). Our static model is built around

four major structural constraints of spatial architecture and dy-namics of forests. First, the allometric relationships (rule 3) in-corporate the essence of tree geometry found at each site thatreflects local environmental conditions (16, 17) and physical con-straints on canopies (21, 22). Second, rule 2 considers average treegrowth and mortality of trees in the most basic way and thereforeincorporates the essence of tree dynamics like recruitment,growth, and mortality. Third, rule 5 that prohibits overlap ofcrowns captures the essence of spatial tree competition in astructural way. Finally, random placement of trees (rule 4) intro-duces the important element of stochasticity that prevents the twotropical forests from being optimally packed. Notably, the in-terplay of these four structural constraints of spatial architecturegenerates the observed tree diameter distributions. Changes in treegeometry and mortality reflect the uniqueness of this interplay(Figs. S3, S8, and S9). We did here a first step in presenting a noveland simple hypothesis for explaining tree diameter distributions.Application of our approach to more forest sites will show if thesimple mechanisms proposed here need to be modified and ifadditional rules are required.Application of stochastic packing theory opens an inspiring new

perspective on the architecture of space filling in forests, but thisapproach is embedded within a long tradition of modeling studies onforest dynamics and self-thinning. For example, the classical −3/2power rule of self-thinning for even-aged forest stands has also beenderived from simple geometric models of space filling together withtree allometries relating diameter to average height, average mass, oraverage crown radius (23). Forest simulation models also make as-sumptions about the geometry and space filling in forests. For ex-ample, several forest models include thinning of the forest in crowdedstands (11). Other studies noted crown plasticity with different crownshapes of canopy trees, which are not anymore positioned above theirstem location as trees grew toward light gaps in the forest canopy (20,24) (Fig. S1C). Surprisingly, our study suggests that competition forlight does not play a direct role in shaping tree diameter distributions,although this is also a central component of models designed to de-scribe the dynamics of vegetation (25, 26). However, rule 5, whichdoes not allow for overlap among tree crowns, and rule 2, whichincludes mortality effects, can be reinterpreted as considering indirectlight effects on tree survival (Fig. 1 B and D).Although irregular sphere packing models in physics maximize

the packing density by rejecting objects (in case of overlappingwith existing ones), the forest crown packing observed is a staticview of forests where trees continuously die and grow. This ap-proach leads to nonoptimal packing where often somewhat tallertrees would fit below a large tree (Fig. 1B). Our sphere packingmodel is static. However, stochastic placement together withcompetition for space (rule 5) captures the outcome of this wellbecause a newly placed tree will in general not have the optimal size

Fig. 3. Observed regional leaf area and tree diameter distributions of the BCIforest and the Sinharaja forest in comparison with results of the forest packingmodel. A and B show results of the forest packing model without tree mor-tality, whereas C–F show results of the model that included mortality. Meanvalues of the model results are shown as solid lines and SDs as polygons orvertical bars (n = 39). We also show the prior of the tree diameter distributionp(d) as gray dashed line (B, D, and F; see model description for details).

Fig. 4. Comparison of field data and model simulations of the distribution ofcrown packing density φ at 20 × 20-m subplots. (A) Results for BCI (n = 1,250) and(B) results for Sinharaja (n = 625). Green solid lines show results of the FPMincluding mortality (mean of n = 39 simulations, polygons representing the SD).

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that could be squeezed in but will be smaller (Fig. 1B). The effectsof mortality and retarded regrowth are reflected in the packingdensity across height (Fig. 2C) and in the leaf area distribution (Fig.2 D and E), which contrasts to approaches that assume equal spacefilling across tree size classes (27) (Fig. S1). However, our modelcannot describe effects of the mortality of large trees that createcanopy gaps that are then colonized by small trees. Fig. 4A showsthat this leads only to a slight underestimation of 20 × 20-m sub-plots at BCI with low packing density.Our tree allometries include the geometric constraint that trees

of a given size can only be filled into a given height layer and thatlarge tree crowns cannot occur near the forest floor. This geo-metric constraint explains partly the low crown packing density ofthe two analyzed forests compared with physical packing systems.Additionally, more than half of the nonoccupied space (∼57%) isfound above 30-m height on BCI. This large proportion is becausethe majority of trees in the forests, whether simulated by ourpacking model or derived from inventory data, are much smallerthan our assumed maximum tree height (of 60 m; Fig. 2B) andonly a small number of emergent trees exceed heights of 30–50 m(1, 28). This finding also contributes to the old debate on strati-fications and layers in tropical forests (29, 30).Our findings provide important insights into the heterogeneous

structure of forests that might be useful also for remote sensing offorests (28, 31). Lidar scanning technologies send beams throughthe canopy and measure the forest structure (e.g., leaf and branchdensity as the emitted beams interact with them) (32). Hereby, thedensity of signals returned at a specific height may be comparableto the observed pattern of leaf area density across height (31).Knowledge on the heterogeneous crown structure of forests de-rived from this study can potentially enhance the interpretation ofsuch remote sensing measurements.

MethodsStudy Areas. For this study, we use forest inventory data on two tropicalforests: one in Panama (BCI; 50 ha) (33–35) and one in Sri Lanka (Sinharaja;25 ha) (36). The plot on BCI is located within the Panama Canal and repre-sents a lowland tropical forest with an annual rainfall of 2,570 mm (1). Theplot in Sinharaja is located in Sri Lanka and constitutes an everwet lowlandforest with an annual rainfall of 5,016 mm on average (1, 36).

The 50-ha plot on BCI hosts ∼230,000 trees at least 1 cm in diameter atbreast height, belonging to more than 300 species (1). The size of the forestplot in Sri Lanka is half that of the forest plot in Panama (i.e., 25 ha), with∼180,000 trees and 209 species (1). Each tree has been identified to speciesand tagged, and its location on the forest plot and stem diameter measured.For both forest plots, the topography is recorded at discrete 5-m steps.

Creating Crown Packings from Forest Inventories. Forest inventories traditionallyinclude information on stem diameter and species, but are increasingly supple-mented by the location of trees (33–36). In this study, we reconstruct the packingof tree crowns by using data on stem location and stem diameter at breastheight. Additionally, relationships between stem diameter d (cm) and other treecharacteristics [like tree height h (m) or crown radius c (m)] are used (18, 19).

We analyzed forest inventories at BCI (census 2010) in Panama and Sin-haraja (census 2001) in Sri Lanka. For both forests, we exclude tree mea-surements if (i) the position of a tree is not within the forest area, (ii) themeasured stem diameter is missing or zero, or (iii) a tree has been declareddead. Stem diameter measurements, which were not taken at breast height(1.30 m above the forest floor), are not corrected (i.e., we assume cylindricalstems for simplification). We assume a maximum tree height of 60 m (1). The

few trees whose height exceeded the assumed maximum height have beenexcluded (tree height is derived from the stem diameter measurements).

As allometric relationships between tree height h(d) (m), as well as crownradius c(d) (m) and stem diameter d (cm), we used for Panama, relationships fromthe literature [i.e., c(d) = 0.37d0.67 and h(d) = 2.74d0.6] (18). Site-specific allometricrelationships for tree geometry are not available for the Sinharaja forest. Wetherefore use allometric relationships for the Dipterocarpaceae family [i.e., c(d) =0.4d0.66 and h(d) = 2.78d0.69] (19). This family shows the highest abundance ofindividuals, number of tree species, and basal area at Sinharaja (1, 36).

Parameterization of the Forest Packing Model. For BCI, tree diameters be-tween dmin = 1 cm (the minimal size in the census) and a maximal dmax = 1.7m has been considered in the forest packing model (for Sinharaja, we useddmin = 1 cm and dmax = 0.86 m). Average tree mortality rates of mBCI = 2.2%/y(for BCI) and mS = 1.3%/y (for Sinharaja) were both determined on the basisof the most recent census data (BCI, 2005 and 2010; Sinharaja, 1996 and2001). To keep the model simple, we assumed constant growth rates typicalfor tropical forests (37) of 0.5 (BCI) (1) and 0.3 cm/y (Sinharaja) (38).

Analysis Tools for Crown Packed Forests. Based on the crown packing system,either created from forest inventory data or simulated using the forestpackingmodel, we calculated howmuch space is filled (crown packing densityφ). For this purpose, a cuboid representing the border of the analyzed forestis used. This cuboid is defined by the ground area and the maximum possibletree height of 60 m (for both forests; SI Methods).

We further divided the forest into distinct height layers to calculate the crownpackingdensity and the leaf areadistribution.Wepresumed linear binning (using1-m layer widths). Logarithmic binned height layers are also possible, but ourresults are independent of the chosen discretization for the height layers (Fig. S2A and B). For each height layer, the leaf area of each tree crown part reachingthat height layer is summed up and divided by the volume of the specific layer(whereby we assume 1 m2 of leaf area per 1 m3 of tree crown volume).

Sensitivity Analyses. We investigated whether the congruence of simulatedforests (using the forest packing model) and the observed forest inventories(exemplary for BCI) would change if the assumed input parameters vary: (i) inminimum and maximum tree sizes (Fig. S3), (ii) in tree geometry relationships(i.e., allometries; Fig. S8), (iii) in the mortality of trees (Fig. S9A), and (iv) in thestopping rule of the forest packing model (Fig. S9B).

Assumptions concerning (i) the crown shape (e.g., spheres or cylinder; Fig.S2C) and (ii) the distribution of leaf area within the tree crown (e.g., fullsphere vs. upper semisphere; Fig. S2D) were additionally examined.

ACKNOWLEDGMENTS. We thank especially two anonymous reviewers,R. D. Holt, J. Brown, and D. Coomes for their helpful comments. We thankN. Gunatilleke and S. Gunatilleke for their support and cooperation concerningthe Ecological Research Project at Sinharaja World Heritage Site. The 25-haLong-Term Ecological Research Project at Sinharaja World Heritage Site is acollaborative project of the University of Peradeniya, the Center for TropicalForest Science of the Smithsonian Tropical Research Institute, and the ArnoldArboretum of Harvard University, with supplementary funding receivedfrom the John D. and Catherine T. MacArthur Foundation, the NationalInstitute for Environmental Science, Japan, and the Helmholtz Centre forEnvironmental Research-UFZ for past censuses. The principal investigators ofthe Research Project at Sinharaja World Heritage Site gratefully acknowl-edge the Forest Department and the Post-Graduate Institute of Science atthe University of Peradeniya for supporting the project, and the local field andlaboratory staff who tirelessly contributed in the repeated censuses of the plot.The Barro Colorado Island forest dynamics research project was founded byS. P. Hubbell and R. B. Foster and is now managed by R. Condit, S. Lao, andR. Perez under the Center for Tropical Forest Science and the SmithsonianTropical Research in Panama. Numerous organizations have provided funding,principally the US National Science Foundation, and hundreds of field workershave contributed. A.H., T.W., and F.T. were supported by an Advanced Grantof the European Research Council (ERC) (233066) and by the Helmholtz AllianzRemote Sensing and Earth System Dynamics.

1. Losos EC, Leigh EG (2004) Tropical Forest Diversity and Dynamism: Findings from aLarge-Scale Plot Network (Univ of Chicago Press, Chicago).

2. CarsonW, Schnitzer S (2008) Tropical Forest Community Ecology. (Wiley-Blackwell, NewYork).3. Ashton P (2014) On the Forests of Tropical Asia: Lest the Memory Fade (Royal Bo-

tanical Garden Kew, Surrey, UK).4. Leigh EGjr (1999) Tropical Forest Ecology (Oxford Univ Press, New York).5. Clark DB, Olivas PC, Oberbauer SF, Clark DA, Ryan MG (2008) First direct landscape-

scale measurement of tropical rain forest Leaf Area Index, a key driver of globalprimary productivity. Ecol Lett 11(2):163–172.

6. Gove JH, Patil GP (1998) Modeling the basal area-size distribution of forest stands: Acompatible approach. For Sci 44(2):285–297.

7. Chave J, et al. (2003) Spatial and temporal variation of biomass in a tropical forest:Results from a large census plot in Panama. J Ecol 91(2):240–252.

8. Coomes DA, Allen RB (2007) Mortality and tree-size distributions in natural mixed-ageforests. J Ecol 95(1):27–40.

9. Whitmore TC (1998)An Introduction to Tropical Rain Forests (Oxford Univ Press, Oxford, UK).10. Moorcroft PR, Hurtt GC, Pacala SW (2001) A method for scaling vegetation dynamics:

The ecosystem demography model (ED). Ecol Monogr 71(4):557–585.11. Bugmann H (2001) A review of forest gap models. Clim Change 51(3-4):259–305.12. Muller-Landau HC, et al. (2006) Comparing tropical forest tree size distributions

with the predictions of metabolic ecology and equilibrium models. Ecol Lett 9(5):589–602.

15128 | www.pnas.org/cgi/doi/10.1073/pnas.1513417112 Taubert et al.

Dow

nloa

ded

by g

uest

on

Sep

tem

ber

26, 2

020

13. Sato H, Itoh A, Kohyama T (2007) SEIB-DGVM: A new dynamic global vegetation modelusing a spatially explicit individual-based approach. Ecol Modell 200(3-4):279–307.

14. Song C, Wang P, Makse HA (2008) A phase diagram for jammed matter. Nature453(7195):629–632.

15. Chiu SN, Stoyan D, Kendall W, Mecke J (2013) Stochastic Geometry and Its Applications(Wiley, New York).

16. Whittaker RH (1956) Vegetation of the great smoky mountains. Ecol Monogr 26(1):1–80.17. Webb LJ (1959) A physiognomic classification of Australian rain forests. J Ecol

47(3):551–570.18. Bohlman S, O’Brien S (2006) Allometry, adult stature and regeneration requirement

of 65 tree species on Barro Colorado Island, Panama. J Trop Ecol 22(2):123–136.19. Kohyama T, et al. (2003) Tree species differentiation in growth, recruitment and allometry

in relation to maximum height in a Bornean mixed dipterocarp forest. J Ecol 91(5):797–806.20. Strigul N, et al. (2008) Scaling from trees to forests: Tractable macroscopic equations

for forest dynamics. Ecol Monogr 78(4):523–545.21. Givnish TJ (1986) Biomechanical constraints on self-thinning in plant populations.

J Theor Biol 119(2):139–146.22. Givnish TJ, et al. (2014) Determinants of maximum tree height in Eucalyptus species

along a rainfall gradient in Victoria, Australia. Ecology 95(11):2991–3007.23. Weller DE (1987) A reevaluation of the -3/2 power rule of plant self-thinning. Ecol

Monogr 57(1):23–43.24. Adams TP, Purves DW, Pacala SW (2007) Understanding height-structured competi-

tion in forests: Is there an R* for light? Proc R Soc Lond B 274(1628):3039–3047.25. Shugart HH (1998) Terrestrial Ecosystems in Changing Environments (Cambridge Univ

Press, Cambridge, UK).26. Landsberg JJ, Waring RH (1997) A generalised model of forest productivity using

simplified concepts of radiation-use efficiency, carbon balance and partitioning. ForEcol Manage 95(3):209–228.

27. Enquist BJ, Niklas KJ (2001) Invariant scaling relations across tree-dominated com-munities. Nature 410(6829):655–660.

28. Morton DC, et al. (2014) Amazon forests maintain consistent canopy structure andgreenness during the dry season. Nature 506(7487):221–224.

29. Richards PW, Walsh RPD (2004) The Tropical Rain Forest (Cambridge Univ Press,Cambridge, UK).

30. Baker PJ, Wilson JS (2000) A quantitative technique for the identification of can-opy stratification in tropical and temperate forests. For Ecol Manage 127(1-3):77–86.

31. Jubanski J, et al. (2013) Detection of large above-ground biomass variability in low-land forest ecosystems by airborne LiDAR. Biogeosciences 10(6):3917–3930.

32. Drake JB, et al. (2002) Estimation of tropical forest structural characteristics usinglarge-footprint lidar. Remote Sens Environ 79(2-3):305–319.

33. Hubbell SP, et al.; de Lao SL (1999) Light-gap disturbances, recruitment limitation, andtree diversity in a neotropical forest. Science 283(5401):554–557.

34. Hubbell SP, Condit R, Foster RB (2005) Barro Colorado forest census plot data.Available at ctfs.arnarb.harvard.edu/webatlas/datasets/bci. Accessed January 16, 2015.

35. Condit R (1998) Tropical Forest Census Plots (Springer, Berlin).36. Gunatilleke CVS, Gunatilleke IAUN, Ethugala AUK, Esufali S (2004) Ecology of Sinharaja

Rain Forest and the Forest Dynamics Plot in Sri Lanka’s Natural World Heritage Site(WHT Publications, Colombo, Sri Lanka).

37. Lamprecht H (1989) Silviculture in the Tropics: Tropical Forest Ecosystems and TheirTree Species: Possibilities and Methods for Their Long-Term Utilization (DeutscheGesellschaft für Technische Zusammenarbeit, Eschborn, Germany).

38. Manokaran N, Kochummen KM (1987) Recruitment, growth and mortality of treespecies in a lowland dipterocarp forest in Peninsular Malaysia. J Trop Ecol 3(4):315–330.

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