A forest structure model that determines crown layers and partitions growth and mortality rates for landscape-scale applications of tropical forests


Correspondence author. E-mail: sbohlman@ufl.edu


1. We present a model to quantify tropical forest structure and explain variance in dynamic rates (growth and mortality) that is computationally simple and can be applied to landscape-scale forest inventory and, potentially, remote sensing-derived data.

2. The model is a modification of the perfect plasticity approximation (PPA) based on tree allometry, tree locations and sizes. The model quantifies crown area index (CAI) (number of crowns per unit ground area) and assigns trees to crown layers, which determines the expected number of crowns above each tree and thus its light environment.

3. The structural model, parameterized and tested for the Barro Colorado Island, Panama 50-ha forest dynamics plot using data from forest inventories and stereo aerial photographs, reproduces most canopy and understorey structural and dynamic properties. The PPA model worked as well or better than a computationally intensive, spatially explicit model. A single allometry for all trees worked equally well as functional group or species allometries. Models of growth and mortality were always improved by adding crown layers as defined by the PPA model.

4. The mean CAI of the 50-ha plot was 3.1 with low variance. The observed variance was lower than when tree locations were randomized, which drastically lowered the variance in tree density per plot, indicating that there are regulating forces towards a small range of crown area indices.

5.Synthesis. A number of simplifying characteristics in structure were uncovered with the PPA structural model applied to a tropical forest: species allometries were not needed despite the high species diversity in the forest; the model worked on a range of plot sizes; and the variance in CAI was surprisingly low, suggesting regulatory mechanisms. The PPA structural model can be used to develop a fully dynamic simulation model for tropical forests. The ability of the simulation model to predict temporal changes in landscape patterns of biomass, dynamic rates, and species and/or functional group composition will provide validation for the partitioning of dynamic rates by crown layers in the PPA structural model.


A challenge in modelling the structure and dynamics of tropical forests is how to make them amenable to predictive modelling on a landscape scale. Growth and mortality, two key processes, are often modelled as a function of the availability of resources, particularly light, which in turn is affected by competition among neighbouring trees. Light profiles in tropical forests have both vertical complexity, due to multiple tree and liana crown layers (Kitajima, Mulkey & Wright 2005; Parker et al. 2005), and horizontal variation, resulting from heterogeneity in forest height and structure (Fernandez & Fetcher 1991; Montgomery & Chazdon 2001). Modelling light availability for tropical trees or predicting it from available data – usually just stem sizes and locations from forest inventories – is difficult because of this vertical and horizontal complexity.

To mechanistically incorporate the dependence of growth and mortality rates on light, a dynamic model must simulate light penetration through the canopy, which depends on estimating the locations and transmissivities of crowns. Spatially explicit models that require the exact location of every tree (Pacala et al. 1996; Chave 1999) are not computationally feasible on a landscape scale. Also, using less detailed properties of forest structure to predict light–growth relationships may provide more stable results than fine-scale modelling of light regimes (Deutschman, Levin & Pacala 1999).

An alternative way of determining dynamic rates (growth and mortality) is through neighbourhood competition indices, which do not measure light effects on dynamic rates directly, but instead determine dynamic rates based on the size and/or distance of neighbouring trees (Gourlet-Fleury & Houllier 2000; Uriarte et al. 2004a,b). Stand simulation models based on competition indices (SELVA: Gourlet-Fleury & Houllier 2000; Gourlet-Fleury et al. 2005; SYMFOR: Phillips et al. 2003, 2004) are not mechanistic however and thus cannot easily predict how forest dynamics respond to variation in climate variables, such as solar radiation, temperature and rainfall. Also, many of these models are spatially explicit and computationally intensive.

Another approach used in some models (FORMIND: Köhler & Huth 1998; Köhler et al. 2001) is to define crown layers that represent cohorts of trees in similar light environments based on fixed vertical height boundaries between layers across the landscape. This approach is similar to the many studies that analyse growth and mortality rates based on fixed diameter or height classes (Condit et al. 2004; Feeley et al. 2007). There is significant evidence for the existence of crown layers (Clark et al. 2008; but see Baker & Wilson 2000). However, layers defined by fixed height boundaries across the landscape cannot directly account for the fact that a tree’s light environment is determined not only by its height but also by the density and heights of neighbouring trees. In this case, density dependence of growth and mortality has to be imposed by other mechanisms in the model (Köhler & Huth 1998; Köhler et al. 2001).

The perfect plasticity approximation (PPA) model (Adams, Purves & Pacala 2007; Purves et al. 2008; Strigul et al. 2008), a descendent of the TASS model (Mitchell 1975; Di Lucca 1998), provides a method of dynamic, rather than fixed height, canopy layer partitioning where layer boundaries are based on local tree densities and sizes. It can be used to identify layers of trees with similar light environments and dynamic rates to simulate height-structured dynamics of forest canopies. In temperate forests, where the understorey is relatively sparse and species diversity relatively poor compared with tropical forests, this simple layering scheme approximates the structure and dynamics of a detailed, spatially explicit model that allows realistic and complex crown displacement and tessellation (Strigul et al. 2008). Here, we present the first test of how well the PPA model can reproduce the features of a structurally diverse, species-rich tropical forest.

We tested three sets of specific hypotheses: (i) The PPA model can predict structural properties of the canopy and understorey layers as well as a spatially explicit canopy structure model. We predict that the allometric relationships used in the model will provide better results if based on species or functional groups, rather than a single relationship for all trees. We hypothesize that there is a narrow range of subplot sizes that optimize predictions. (ii) PPA layers explain significant variation in dynamic rates independent of tree size. (iii) The variance in the number of crown layers is high, as it is expected to vary with forest height and with different habitat types. We also compare the number of layers predicted for a forest in Panama to empirical data on layering in a nearby forest in Costa Rica.

Materials and Methods

The data for this study were collected on Barro Colorado Island (BCI) (9°09′ N, 79°51′ W), a 15-km2 island covered with semi-deciduous lowland moist forest located in Lake Gatun in the Panama Canal (Dietrich, Windsor & Dunne 1982). The old-growth forest on the island is believed to have been mostly free of agricultural clearing for the past 1500 years and has a minimum age of 500 years (Piperno 1990). The island receives an average of 2650 mm year−1 of rainfall. Between 1980 and 1982, a 50-ha permanent plot was established in which every tree stem ≥ 1 cm d.b.h. (diameter at breast height) was mapped and identified to species level, and its d.b.h. was measured. Every 5 years since 1980, the d.b.h. of all trees has been re-measured and tree mortality and recruitment into the 1-cm d.b.h. class were recorded. In 2000, there were 303 species on the plot. Detailed descriptions of BCI and the 50-ha plot can be found elsewhere (Hubbell & Foster 1983; Leigh 1999).

Model Description

The study area’s crown area index (CAI) (number of crowns per ground area) and each tree’s crown layer occupancy were determined using the crown model module of the PPA model as described by Purves et al. (2008) and Strigul et al. (2008). The PPA model is an individual tree-based forest dynamics simulator where cohorts of trees grow, die and are recruited into the forest. The crown model module assigns trees into either the canopy or understorey crown layer, which determines the cohorts’ dynamic rates. The module requires the height and crown area of every tree and a crown shape. In the simplest version of this model, which was used here, the crown shapes are flat discs with heights and crown areas determined by allometric relationships with the trees’ diameters.

The assignment of crowns to the canopy or understorey is determined by first sorting all trees from a plot from tallest to shortest. Starting from the tallest, the crown area of each successive tree is assigned to the canopy layer until the cumulative canopy crown area fills the whole plot, thus creating a single full layer. The last tree to ‘enter’ the canopy layer, i.e. the shortest tree in the canopy layer, defines the height, which we term inline image, above which all trees are in the canopy and below which all trees are in the understorey. The sum of the crown areas of all trees taller than inline image equals the plot area. The PPA model is spatially implicit; the exact location of individual trees within a plot does not affect the outcome of the model. All canopy layer trees are assumed to receive full sun even if their stems are highly clumped within a plot. Thus, there is an assumption of a high degree of potential displacement of crowns from trees stems in this model.

An important modification of the PPA crown model for tropical forests was the addition of multiple understorey crown layers. In analyses of temperate forest inventory data, where the PPA crown model has thus far been applied, the cumulative crown area of the understorey was less than the total plot area. For BCI, the cumulative understorey crown area greatly exceeded the total plot area so additional ‘understorey layers’ were added. It is important to note that we do not believe these understorey layers are continuous coherent layers. Instead, each understorey layer defines a collection of trees above which we can expect to find a certain number of crowns. For example, above trees in the tallest understorey ‘layer’, we can expect to find on average one tree crown. Trees were assigned to the tallest understorey layer until the cumulative crown area of this layer filled 100% of the plot area. The shortest tree to enter the 1st understorey layer determined the inline image height, which is the boundary between the 1st and 2nd understorey layers. Trees shorter than inline image filled the 2nd understorey layer until it was filled to 100% and generated a inline image height boundary between the 2nd and 3rd understorey layers and so on.

The 50-ha plot was subdivided into subplots that ranged in size from 0.015 to 25 ha, and the crown model was implemented for each subplot. These subplots defined the neighbourhood of interaction among the trees and allowed the Z* values to vary spatially. Each tree’s crown was assigned entirely to the subplot where the stem was located, although in reality part of the crown may be located in an adjacent subplot. Input data were the assignment of trees to subplots and their diameters, determined from the 50-ha forest inventory, and diameter-based allometries that predict height and crown area, derived from both field and remote sensing data. The crown layer occupancy of each tree was predicted from the model and compared against field observations, but these field observations were not used to parameterize the model.

To compare to the PPA results, we also determined each tree’s crown layer using a spatially explicit canopy structure model in which tree crowns were assumed to be directly above the tree stems with no crown displacement (‘rigid crown model’) (Pacala et al. 1996). In this model, the exact location of each tree in relation to other trees determined whether it is in the canopy or understorey. Crown areas and tree heights were determined using a single allometric equation applied to all trees. Crowns were modelled as circles with the centre directly above the stem and a radius determined from the diameter-crown area allometry. A 0.25-m grid was placed over the 50-ha plot. We determined the tallest tree crown overlapping each grid point and assigned it as the canopy tree at that point. Crowns that were not the tallest tree at any grid point were considered understorey trees. Like the PPA model, this model was applied to subplots with a width of 31.25 m. Only trees within the subplot ‘competed’ to be the tallest tree at each point in the subplot. If a crown was the tallest tree for ≤ 4 grid points [a sun-exposed crown area (ECA) ≤ 1 m2, which was the smallest crown area detected in the aerial photographs], it was considered to be in the understorey.

Crown Map from Aerial Photographs

High-resolution stereo aerial photographs were taken in September 2000. Targets within the plot served as ground control points and allowed us to translate measurements of the photographs into coordinates of the plot. Each crown’s outer boundary and maximum height in 10 ha of the plot were digitized. Between 2002 and 2005, images of the 1:6000 photographs overlaid by digitized crown boundaries were taken into the field to determine which tagged tree stems linked to which digitized crowns, generating a canopy layer crown map with each crown identified with a tag, species and diameter taken from the stem database. Further methods are in Appendix S1 in Supporting Information.

Allometric Relationships

We developed two types of diameter-crown area allometries: ECA and total crown area (TCA). ECA, which is only the part of the crown visible from directly above and does not include shaded portions of the crown, was determined from the digitized crown boundaries from the photographs. TCA, which includes the full extent of each crown, both sun-exposed and shaded, was determined from ground measurements. Diameter-height and diameter-TCA allometries were developed from field measurements reported in Bohlman & O’Brien (2006), and data provided by H. Muller-Landau for BCI and by J. Wright for a nearby plot on the Gigante Peninsula. Allometric relationships were determined for all trees together (‘generic’) and for functional groups and species that had ≥ 6 observations. Species were classified into two sets of functional groups (Poorter 2007): adult stature, consisting of tall trees, mid-sized trees, treelets and shrubs, and regeneration requirement, consisting of light-demanding, intermediate and shade-tolerant trees (Condit, Hubbell & Foster 1996; Bohlman & O’Brien 2006). A separate allometric equation was developed for each functional group and combinations of the two sets of functional groups (for example, tall, light-demanding species). Additional methods on allometries are in Appendix S1.

Model Runs and Diagnostics

The PPA crown model was applied to forest inventory data from the years 1995 and 2000 varying several factors: which type of crown area (ECA or TCA) was used to fill the top crown layer; whether generic, functional group or species allometry was used for all layers; and subplot size. When ECA was used to fill the canopy layer, the canopy layer only was filled to 91%, rather than 100%, to represent intercrown spacing in the canopy layer (see Appendix S1). The subplots were squares with widths of 12, 25, 31.25, 41, 50, 62.5, 83.3, 100, 125, 250 and 500 m. We also used subplot widths of 5 and 10 m to compare the predicted number of crown layers to the number of leaf strata measured by Clark et al. (2008). Clark et al. (2008) destructively harvested all leaves from a vertical column of forest 4.6 m2 in size and determined leaf strata based on alternating vertical sections of high [mean leaf area index (LAI) of 2.0] and low leaf density (mean LAI of 0.18). Our subplot areas (25 and 100 m2) were larger because the PPA crown layer estimates were based on stem locations and a larger footprint was needed to include trees whose foliage would contribute to a column of forest 4.6 m2 in area. We assumed that one leaf stratum determined by Clark et al. (2008) equalled one crown layer of the PPA model.

For each model run, the predicted canopy status of each tree (whether a tree is in the canopy or understorey) was compared with field observations of canopy status determined by a field observer for 2845 trees in 1995 (Wright et al. 2005) and via the aerial photographs for 36 215 trees in 2000. A tree was considered to be in the canopy only if it had any part of its crown visible from directly overhead. We evaluated model predictions for all trees together for 1995 and 2000, but for individual subplots only for 2000 because of limited trees per subplot in 1995. Canopy status was determined for all trees in the 50-ha plot for each model run, but model diagnostics were based only on trees for which canopy status observations were available. We determined how well the model reproduced the observed number of trees in the canopy and understorey spatially (per subplot) and for different size classes (50-mm d.b.h. classes). We determined the inline image height for each subplot that maximized the correct classifications of canopy status and compared this optimal inline image to the inline image predicted by the model. We also analysed predicted and observed functional group composition of the canopy and understorey.

To compare results for different subplot sizes, we quantified the goodness-of-fit between the predicted percentage of trees in the canopy and the observed percentage of trees in the canopy across different d.b.h. classes for different subplot sizes. As a goodness-of-fit test, we quantified the percentage of the variation in the observed percentage of trees in the canopy explained by the predictions as:


where SS is the sum of squares, yi is the mean observed percentage of trees in the canopy per diameter class, inline image is the predicted mean percentage of trees in the canopy per diameter class and inline image is the observed mean among diameter classes.

We also determined the probability of predicting trees in the correct layer (canopy or understorey). We calculated the percentage of trees that were correctly predicted to be in the canopy when they were observed in the canopy and the percentage of trees observed in the canopy when they were predicted in the canopy. This was also carried out for the understorey. For both crown layers, these metrics were also calculated by summing crown areas rather than the number of trees, thus determining the percentage of crown area that was correctly predicted. For different subplot sizes, we also calculated the percentage difference between mean growth and mortality rates for trees observed versus predicted in the canopy and understorey.

Evaluation of Dynamic Rates

To evaluate demographic and population dynamic implications of canopy layering, we compared mean growth and mortality rates of trees predicted and observed in the canopy and understorey. The methods for calculating growth and mortality rates are provided in Appendix S1. Dynamic rates of tropical trees are often calculated for size categories such as saplings, pole-sized trees and adult trees (Feeley et al. 2007). We explored the alternative classification of dynamic rates based on crown layer occupancy as determined by the PPA model. We used maximum likelihood methods to determine whether a layer-based model, a size-based model, or a combined layer- and size-based model better explained the variance in diameter growth and mortality among trees. Eight models were tested for each dynamic rate for the 1995–2000 and 2000–2005 census intervals. Half of the models did not include crown layers [models in column (a) in Table 1], and the other half did [models in column (b) in Table 1]. The diameter categories used for models 2 and 4 were 10–49 mm, 50–99 mm and ≥ 100 mm d.b.h. (Feeley et al. 2007).

Table 1.   AIC values for models of diameter growth and mortality with (column a) and without (column b) the addition of crown layers. Dynamic rates were calculated from 1995 to 2000 and 2000 to 2005. Crown layer designations used a subplot width of 31.25 m and a single allometry for all trees. For models 2 and 4, d.b.h. classes used were 10–49, 50–99 and ≥ 100 mm
Model no.Growth(a) No crown layers(b) Four crown layersDifference(a) No crown layers(b) Four crown layersDifference
1Single value – all trees275 337239 08836 249276 880245 82431 056
2Single value per d.b.h. class234 276231 6132 662242 317239 0433 274
3Function of d.b.h. – all trees242 045230 49211 553250 227239 14011 087
4Function of d.b.h. within d.b.h. class228 820227 931889238 429236 5731 856
Model no.Mortality(a) No crown layers(b) Four crown layersDifference(a) No crown layers(b) Four crown layersDifference
1Single value – all trees203 278202 394884171 328171 019309
2Single value per d.b.h. class202 472202 189284171 168170 973195
3Function of d.b.h. – all trees201 865201 669197170 857170 694163
4Function of d.b.h. within d.b.h. class201 791201 637154170 832170 716116

Growth was modelled using the log normal distribution for errors (Condit et al. 2006) with standard deviation modelled separately for each d.b.h. category or crown layer. For the growth model analysis only, growth rates ≤ 0 mm year−1 were set to 0.5 mm year−1 because the lognormal distribution does not allow negative growth rates (Condit et al. 2006). For models 3–4 (Table 1), growth was modelled as a power function of diameter.


where ag and bg are estimated for all trees together (model 3) or each diameter category (model 4).

Mortality for the entire census interval, rather than annual mortality, was used. For models 3–4 (Table 1), mortality was modelled with a functional form to allow for U-shaped curves. The binary mortality data were transformed via the logit function to a continuous distribution that was more easily modelled as follows:


where P is the probability of mortality, and am, bm and cm are estimated either for all trees together (model 3) or each d.b.h. category (model 4). The likelihood of the entire data set was calculated as (Canham, Papaik & Latty 2001):


The different models were compared using Akaike Information Criteria (AIC).

Spatial Patterns of Crown Layers

The mean and standard deviation of the CAI per subplot were calculated for 1995 and 2000 using a subplot width of 31.25 m and species allometry. We calculated the average and standard deviation of CAI for different habitat types (Harms et al. 2001) and used analysis of variance and Tukey’s Honestly Significant Difference tests at = 0.05 level to detect whether any of the habitats had a statistically different CAI. We determined the correlations between CAI, tree density and maximum tree height per subplot and used multiple regression analysis to determine the extent to which tree density and maximum height explained the variation in CAI. To examine whether the variance in CAI was unexpectedly low, we compared the observed variance among subplots to the variance obtained if all stem locations and sizes were randomized for the entire 50-ha plot, which should greatly reduce clumping of stems. If the observed variance is <95% of the 1000 randomizations, this indicates that the observed variance in CAI is lower than the variance produced by a random distribution of stems.

Statistical analyses were carried out with r version 2.9.0 (R Development Core Team 2011), using the ‘bbmle’ package (Bolker 2008).


Predicted Versus Observed Canopy Properties

Despite the simplicity of this model, most predictions of canopy and understorey structural and dynamic properties matched the observed values for 1995 and 2000. The number of trees predicted in the canopy and understorey matched observations for nearly all subplots (Fig. 1), although there is scatter in the relationship for canopy trees. The crown model accurately predicted the percentage of trees in the canopy per diameter class (Fig. 2) and the functional group composition in the canopy and understorey (Fig. 3). The PPA-predicted inline image height threshold was close to the optimal height threshold (the value for each subplot that best predicted canopy status) for most subplots, especially for height thresholds <25 m (Appendix S2). The PPA model fits the observed values better than the rigid crown model. The rigid crown model placed too many trees in the canopy in size classes <550 mm d.b.h. (Fig. 2) for nearly all subplots (Fig. 1).

Figure 1.

 Observed versus predicted number of trees in the canopy (main) and understorey (inset) per subplot in 2000. Only trees with an observation of canopy status were included. There were not observations for all trees in all subplots. Main plot: The lines are from linear regressions for different types of allometries used in the perfect plasticity approximation (PPA) and rigid crown models. The values in parentheses are r2 values. The circles are from the PPA model using a single (generic) allometry for all trees. Inset: All regression lines fall on the 1:1 line except for the rigid crown model (dashed line).

Figure 2.

 Fraction of trees in the canopy for different diameter classes in 2000. The solid line is from the observed data. The thick dashed line is from the rigid crown model predictions. All other thin lines are results predicted from the perfect plasticity approximation model. For the separate dashed line, a single ‘generic’ allometry between diameter and total crown area (TCA) was used for all trees. For all other lines, allometry between diameter and sun-exposed crown area (ECA) was used. The subplot width was 31.25 m.

Figure 3.

 Percentage of functional groups for observed versus predicted canopy (closed symbols) and understorey trees (open symbols) in 2000. Regeneration requirement functional groups (squares) are gap, inter (intermediate) and shade. Adult stature functional groups (circles) are tall, mid (mid-sized), treelet and shrub. Species allometry and a subplot width of 31.25 m were used.

The use of generic, functional group or species allometries made little difference in the results (Figs 1 and 2). However, better results were produced using ECA, rather than TCA, for the canopy layer allometry (Fig. 2). Using TCA allometry caused too few trees predicted to be in the canopy for trees <500 mm d.b.h.. Examination of the ECA and TCA allometries provides context for these results. For all trees together, TCA is greater than ECA for trees with d.b.h. < 800 mm (Fig. 4a), although ECA is greater than TCA for trees with d.b.h. > 800 mm (Appendix S3). Four combined regeneration requirement/adult stature functional groups composed 88% of the canopy trees. For ECA, these four functional groups show some variation, but less than the difference between ECA and TCA (Fig. 4b). The differences in allometry among the functional groups are even smaller for TCA allometry (Fig. 4c). Dbh–height allometries were nearly identical for the most common functional groups (Appendix S3).

Figure 4.

 Allometric relationships between d.b.h. and crown area for all trees together and for the most common combined functional groups in the canopy. (a) Sun-exposed crown area (ECA) and total crown area (TCA) for all trees together; (b) ECA for the most common functional groups and all trees together (dark lines) and TCA for all trees together (grey line). (c) TCA for the most common functional groups and all trees together (dark lines) and ECA for all trees together (grey line). The same figures for the full diameter range are presented in Appendix S3.

Dynamic rates

For trees with observed canopy status only, the average dynamic rates for trees predicted in the canopy and understorey were within 10% of the rates for trees observed in the canopy and understorey, with the exception of canopy mortality in 2000, which was overpredicted by 75% (Table 2). If mortality is based on crown size, the predicted 2000 canopy mortality (1.70% canopy area year−1) is only 25% greater than observed canopy mortality (1.36% canopy area lost year−1), indicating that a disproportionate number of canopy trees predicted to die but observed to survive were among the smallest canopy trees. For both 1995 and 2000, mean growth rate per layer decreased from the canopy to the bottom understorey layer (Table 2). In both years, mortality was similar for the tallest two crown layers and then increased in the lower crown layers (Table 2). Within each layer (i.e. for trees with on average the same number of crowns above them), growth increased with diameter, whereas mortality usually declined up to a d.b.h. of 10 cm (Appendix S4).

Table 2.   Mean and standard deviation of growth and mortality for trees predicted and observed to be in the canopy and understorey (for trees with canopy status observations only*), as well for trees predicted to be in each crown layer (for all trees on the 50-ha plot). The subplot width was 31.25 m, and species allometries were used. Observed growth and mortality rates differed between 1995–2000 and 2000–2005 because the size distribution of trees with measurements differed between the two census intervals
Growth (mm year−1)Mortality (% year−1)d.b.h. (mm)Growth (mm year−1)Mortality (% year−1)d.b.h. (mm)
Observed canopy*6.29 ± 5.931.69 ± 0.070401 ± 2673.52 ± 4.411.18 ± 0.045268 ± 220
Predicted canopy*5.76 ± 5.891.64 ± 0.065390 ± 2573.15 ± 4.162.15 ± 0.061264 ± 211
Observed understorey*2.40 ± 3.401.69 ± 0.078175 ± 1410.665 ± 1.162.57 ± 0.01936 ± 38
Predicted understorey*2.57 ± 3.421.76 ± 0.086155 ± 1360.688 ± 1.272.49 ± 0.01935 ± 37
Predicted layer 1 (canopy)3.13 ± 4.892.20 ± 0.028247 ± 2072.94 ± 4.422.31 ± 0.029251 ± 213
Predicted layer 2 (understorey)1.46 ± 2.572.12 ± 0.022102 ± 791.51 ± 2.442.24 ± 0.023102 ± 82
Predicted layer 3 (understorey)0.656 ± 1.1203.01 ± 0.01132 ± 230.667 ± 1.182.50 ± 0.01033 ± 24
Predicted layer 4 (understorey)0.478 ± 0.7063.81 ± 0.01618 ± 90.457 ± 0.7963.03 ± 0.01519 ± 9

Regardless of how growth and mortality were initially modelled using diameter classes and diameter-based functions (Table 1; Models 1a–4a), the maximum likelihood of the model was always improved significantly by adding crown layers (Models 1b–4b), especially for growth. The difference between corresponding models without crown layers (Models 1a–4a) and with crown layers (Models 1b–4b) was >100 AIC units, indicating the models without crown layers have essentially no statistical support compared to models with crown layers (Burnham & Anderson 2002). The growth and mortality models with greatest support in both 1995 and 200 included both crown layers and diameter-based functions (Table 1).

Variation in subplot size

The optimal range of subplot widths for predicting canopy status and obtaining the correct mean dynamic rates was 25–50 m. This range provided the best fit between observed and predicted percentage of trees in the canopy per d.b.h. class (Fig. 5). At all subplot widths, the probability of predicting the correct trees in the understorey was >95%. However, only for subplot widths between 25 and 50 m was the percentage of trees observed in the canopy when predicted in the canopy and the percentage of trees predicted in the canopy when observed in the canopy ≥ 60% (Fig. 5, Appendix S5). Similarly, the differences in understorey growth and mortality rates were ≤ 10% at all subplot widths. However, only for subplot widths between 31.25 and 50 m was the difference in observed versus predicted canopy growth rates <10% (Appendix S5).

Figure 5.

 Fraction of variation in the observed percentage of trees in the canopy explained by the predicted percentage of trees in the canopy for different subplot widths. The percentage of trees in the canopy was divided by 50-mm-diameter classes (Fig. 2). Functional group allometry was used. The two vertical lines mark the subplot size range determined to be optimal.

Spatial Variation in Canopy Structure

The mean CAI was 3.1 ± 0.169 for 1995 and 2000 (Table 3). CAI was between 2.5 and 3.5 for 77% of the subplots (Appendix S6). The mean maximum height of the forest was 33 m, and the mean lower boundary of the canopy layer (inline image ) was 17 m, indicating the average canopy layer depth was 15 m (Table 3). The variance in CAI among subplots was low. Although observed variance among subplots in tree density was much higher than in the randomizations, the observed variance in CAI was lower than 95% of the randomizations (Table 3). There was little variation in CAI among habitats (Appendix S6). The number of trees and the maximum height per subplot had significant positive correlations with CAI per subplot (Appendix S6). A multiple regression showed these two variables explained 48% of variation in CAI.

Table 3.   Mean and variance in structural characteristics predicted by the perfect plasticity approximation model for the 50-ha plot on Barro Colorado Island, Panama. Z* heights are the heights of the boundaries between crown layers. The range of 95% of the randomizations is given. The subplot width was 31.25 m, and species allometries were used
Structural characteristic19952000
Crown area index – observed3.1190.1693.0740.169
Number of trees ha−1– observed44753064184441
Number of canopy trees ha−127139732634229
Maximum height (m)33.218.533.318.1
inline image(m)17.55917.963.6
inline image(m)10.812.810.913.3
inline image (m)

The mean CAI for the BCI 50-ha plot closely matched the mean number of leaf strata reported by Clark et al. (2008) for La Selva, Costa Rica (Table 4). The coefficient of variations were similar, as well as the strength of the correlation between number of layers/strata and maximum height (Table 4).

Table 4.   Comparison of mean number of crown layers at Barro Colorado Island (BCI), Panama and leaf strata at La Selva, Costa Rica. Number of strata at La Selva was determined by partitioning 4.56-m2 vertical columns of forest into sections of high and low leaf area density through leaf harvesting (Clark et al. 2008). The mean number of leaf strata was determined by averaging the leaf strata values presented in Fig. 5 of Clark et al. (2008). Number of layers at BCI was determined using the PPA model with the 1995 stem map and a single allometry for all trees
LocationSubplot size (m)Mean no. of layersStandard deviation no. of layersCoefficient of variationCorrelation between max height and no. of layers
La Selva52.891.1120.390.67


The PPA crown layer model, despite its simplicity, accurately predicted whether trees were in the canopy or understorey, explained important variation in dynamic rates and gave insight into forest structuring. It provides a framework to define cohorts of trees with similar light environment and dynamic rates that could serve as the basis for landscape-scale dynamic modelling in tropical forests.

Accuracy of The Crown Model

The crown model was successful in predicting subplot-level characteristics of canopy structure. The model reproduced observations from data sets using different methods (aerial photographs versus ground observations) from two separate years (1995 and 2000). The PPA model fit observations better than a spatially explicit rigid crown model, which highlights the importance of crown displacement in structuring canopies. The rigid model designated too many trees, especially small trees, into the canopy. In a forest with clumped tree distributions, competitive interactions among clumped trees of similar heights force crowns to grow and extend away from each other, overtopping nearby smaller trees. These smaller trees will be designated correctly as understorey trees in the PPA model because the tallest trees in a subplot are placed in the canopy, but erroneously designated as canopy trees in the rigid crown model.

At the onset of this research, we assumed the high species diversity and structural complexity of this tropical forest compared with temperate forests would require careful tuning and extra parameterization of the model. In fact, the model proved robust to highly grouped allometries and different subplot sizes and did not require extra parameterization. Although tropical trees show great variation in branching architecture (Hallé, Oldeman & Tomlinson 1978), the variation in overall crown shapes among tropical species may be limited compared with temperate forests, which have both conifers and broad-leaved trees. The fact that ECA allometry had a better fit than TCA allometry suggests that crowns do not conform to the flat-topped crown shape used in the PPA model here. Using a rounded crown shape model that partitions each tree’s crown into sun-exposed and shaded portions (Purves, Lichstein & Pacala 2007) should provide a successful alternative to using an ECA allometry.

The model performed optimally over a fourfold (625–2500 m2) range of subplot areas, which determines the neighbourhood of competition. Given that the BCI forest is a mosaic of tall, mature forest and gaps of various ages, we assumed there would be a narrow range of subplot sizes, particularly the size of an average gap, which would capture homogeneous patches of forest allowing the crown model to work well. In fact, the optimal subplot was much larger than the average gap size on BCI, which is 86 m2 (Brokaw 1982). Instead, the optimal subplot sizes were on the order of the size of 1–3 of the largest tree crowns on BCI (c. 1000 m2) or 20–80 average-sized canopy trees. The optimal subplot size range means some subplots contain a heterogeneous mix of gaps and mature forest. This subplot size may produce good results because average-sized gaps are a small component of subplots and incorrect predictions of canopy status of the gap trees may not produce poor subplot-level diagnostics.

Dynamic Rates

The average dynamic rates for trees predicted in the canopy and understorey matched the averages for trees observed in the canopy and understorey in most cases. Correct representation of dynamic rates for different canopy layers is important for using the forest crown model in dynamic modelling. The exception was canopy mortality in 2000, for which predicted canopy mortality was greater than observed canopy mortality. There are several possible explanations for the discrepancy. Trees that were predicted to be in the canopy but were actually in the understorey may be receiving too little light, given their size and neighbourhood, and were more susceptible to mortality. Second, the observed canopy mortality rate could be artificially low due to field observation errors. Several years elapsed from the time the aerial photographs were taken to when the trees mapped in the photographs were linked to stems on the ground. During this time, some canopy trees died. We made an effort to either identify the canopy trees that had died in treefall gaps or exclude these areas from analysis. But there may be some cases where trees that had been in the understorey when the photographs were taken were mistakenly identified as canopy trees in the field as they had replaced a canopy tree that had died.

Division of dynamic rates by crown layer captured some of the variation in growth and mortality but did not supplant the variation in dynamic rates related to diameter. The best models of dynamic rates included both crown layer- and diameter-based differences. The few empirical studies (Davies 2001; Metcalf et al. 2009; Rüger et al. 2011) that separate the effects of light exposure and size on the tropical tree dynamic rates find that size had an effect on dynamic rates of nearly all species, but light had a less consistent effect across species, especially for mortality. Given the modest difference in growth and mortality observed among layers at the same diameter in many cases (Appendix S4), one might question whether including crown layers in a dynamic model is necessary, rather than determining dynamic rates as a function of diameter alone. A crucial difference of using a crown layer- versus a diameter-based model is that structuring the forest with crown layers provides an implicit mechanism for competition and density dependence that a purely diameter-based model lacks because the crown layer assignment of a tree, and thus its dynamic rates, depends on the size and density of trees within the subplot. Other forest models must include density-dependent changes in dynamic rates through separate processes, for example, by directly estimating the increased mortality due to crowding or by decreasing growth with depth in the canopy profile via explicit light competition and photosynthesis modelling (Köhler et al. 2001).

Regulation of Crown Layers in the Forest

The mean and variance in CAI for BCI and number of leaf strata at La Selva, Costa Rica (Clark et al. 2008) were similar. This lends support for the PPA CAI model, and this is a much simpler method to determine strata than intensive ground measurements used by Clark et al. (2008). The spatial variability in CAI and leaf strata was surprisingly low, suggesting regulation in the vertical structuring of the forest towards a consistent number of crown layers, which may be due to resource limitation, particularly for light. The CAI may not go above three because the light levels in a fourth layer of vegetation (where a tree would have three crowns above) may be too low for the survival of even shade-tolerant trees. On the other hand, a forest stand does not stay below three layers for long because shade-tolerant trees that can tolerate the light levels of the second and third crown layers can recruit to the stand and because forest growth rates are high.

Forest Modelling

The PPA structural model presented here provides an important first step in developing a fully dynamic tropical forest PPA simulation model (Purves et al. 2008). Information on how well this simulation model can reproduce temporal and spatial patterns of biomass, dynamic rates and species/functional group composition is necessary to validate the results of this model, particularly the partitioning trees into crown layers and assigning dynamic rates. Although the structural model based on a single allometry for all trees performed well, we expect variations in dynamic rates related to functional groups or species will be necessary to reproduce essential patterns with the simulation model.

The PPA model adds to the diversity of modelling approaches available to address different types and scales of questions for tropical forests. Because the PPA model is spatially implicit and assumes infinite crown displacement within a given subplot, the PPA model may be less useful than spatially explicit models (Phillips et al. 2003, 2004; Gourlet-Fleury et al. 2005) for applications, which require tracking individual trees and describing reaction to small-scale disturbance, such as selective logging. The tropical forest PPA model is computationally efficient and accurately describes landscape variation in the vertical structure. It is particularly well-suited for landscape-level simulations, for example simulating broad-scale patterns of biomass and species turnover during succession and over environmental gradients (Purves et al. 2008). Forest inventory plot networks exist in many tropical regions (e.g. Peacock et al. 2007; Lewis et al. 2009) to which the PPA model can be applied to examine landscape-scale patterns. Furthermore, high spatial resolution remote sensing data provide the possibility of obtaining continuous, landscape-scale quantitative data on canopy layer trees (Palace et al. 2008; Barbier et al. 2010) for use in the PPA model.

The tropical forest PPA model can potentially be used for theoretical analysis similar to the temperate forest PPA model, which has been used to predict community-level properties such as early- and late-successional dominant species and root–shoot–stem allocation over soil nutrient gradients (Purves et al. 2008; Dybzinski et al. 2011). To do so with the tropical forest PPA will require accounting for the multilayer structure and other features of tropical forests.


Drew Purves assisted in the initial model development. Richard Grotefendt of Grotefendt Photogrammetric Services took and processed the stereo photographs and assisted with crown digitizing. Andrew Hida provided much of the field work for the stereo photographs. We thank Helene Muller-Landau and Joe Wright for providing allometric and canopy status data. S.A.B. was supported by the Carbon Mitigation Initiative (CMI) of the Princeton Environmental Institute at Princeton University. CMI (http://cmi.princeton.edu/) is sponsored by BP and Ford. The stereo photograph collection, field-truthing and analysis were supported a NASA Earth Systems Science Fellowship and Center for Tropical Forest Science (CTFS) research grant. The BCI forest dynamics research project was made possible by National Science Foundation grants to Stephen P. Hubbell, support from the Center for Tropical Forest Science, the Smithsonian Tropical Research Institute, the John D. and Catherine T. MacArthur Foundation, the Mellon Foundation, the Small World Institute Fund, and numerous private individuals, and through the hard work of over 100 people from 10 countries over the past two decades. The plot project is part CTFS, a global network of large-scale demographic tree plots.