Previous modeling approaches

To understand the interplay between forest structure and local or regional climate it is necessary to both understand the competitive dynamics of trees within a stand and to couple tree physiology – at the individual or whole forest scale – to environmental conditions. At the regional scale, a common approach has been to focus on vegetation types coupled to atmospheric conditions. These models have successfully captured the geographic distribution of vegetation types and net primary productivity as well as environmental processes such as moisture flux and runoff [7]–[9], [31]–[35]. For understanding fine-scale forest structure explicit temporal and spatial modeling and simulation efforts are becoming increasingly accurate at capturing local forest dynamics. Several models which aim to capture local phenomena focus on the small-scale competition of trees represented either as components or patches (e.g. the JABOWA model [17], [18]) or explicitly as individually trees (e.g. SORTIE [16], [17], [19] and TASS [20]). These models predict the gap structure of the canopy [17]–[19], the species composition and diversity of a stand [16], [18], [19], the standing biomass [16]–[20], and the size distribution of trees [20] at the local scale. In the case of SORTIE, the model tracks individual trees and simulates the coupled dynamics of canopy spatial structure, crown competition, light availability, seedling recruitment, growth, and tree mortality [16], [19].

The drawback of these models is that they are computationally expensive when applied to larger regions. The more recent efforts of the perfect plasticity approximation (PPA) have used basic assumptions about the interaction of individual trees to produce macroscopic equations (analogous to those found in statistical physics) for features such as the equilibrium size distribution of trees [14]. This technique captures the average interaction between competing trees without explicitly modeling each individual and thus can be inexpensively applied to larger regions. For features which represent the average of numerous trees (e.g. total density and average height) the PPA produces very similar results to the models which explicitly track individual trees [14]. The PPA also compares well with measurements for crown characteristics such as depth and radius [11] and the temporal dynamics of stand structure, biomass and successional patterns [15].

The models discussed thus far focus primarily on either the competition dynamics within a stand or the regional coupling of environmental conditions to vegetation. The ecosystem demography model (ED) connects these two approaches in an effort to more accurately understand forest dynamics coupled to the environment at multiple scales [12]. ED relies on plant functional types as a means for capturing local forest diversity and, similar to the PPA discussed above, relates an ensemble average to the complex dynamics of individual trees including the stochastic processes of mortality and succession. ED then couples this ensemble approach with numerous environmental processes such as atmospheric conditions, fire, evapotranspiration, and carbon sequestration. ED is able to capture important local and regional phenomena such as carbon flux, standing biomass, the stock of soil carbon, or the response of productivity to changing climate [4], [5], [12], [13]. Approaches like this hold much future promise for understanding both small-scale forest structure and regional vegetation patterns as they feedback with climate. However, these models require explicit temporal simulation, and decisions about how to represent plant diversity and physiology.

Steady-state allometric approach

Distinct from the models discussed above, the framework that we develop in this paper consists of a steady-state analytic calculation rather than a temporally and/or spatially explicit simulation. Our framework takes average local meteorology as an input and numerically calculates maximum tree height as an output. In comparison with the models discussed above our framework is not able to characterize detailed local phenomena such as temporal dynamics or species composition, but it does allow us to understand the average tendencies and constraints facing trees across different environments and this provides a useful foundation for incorporating more complicated processes.

We employ a single generalized tree across a range of environments without specific knowledge of local plant functional types commonly used in previous models [7]–[9], [34], [35]. In doing so we sacrifice accuracy at the local scale but gain a simple understanding of the average variations across environments. In the context of resources our framework lends insight into the mechanisms underlying deviations from the allometric scaling laws where, for example, we are able to show how different tree traits are suited to a given environment and predict the temperature-based variation of stomatal density (Fig. 5). These variations in turn modify the size-based scalings for an individual tree species (see Supplement S1). Ongoing work is beginning to understand departures from the basic zeroth order allometric scaling laws [24], [27], [36], [37], however, it should be noted that the zeroth order theory has yet to be coupled with environments in order to test its predictive power. Our work provides another means for expanding the basic allometric scaling laws to encompass features that are relevant to more detailed modeling efforts.

Ecological relevance of tree height

We demonstrate the utility of our framework by predicting maximum tree height. We choose to focus on tree height because size is a natural quantity within the allometric framework and because height is an important indicator of various consequential features of a forest, such as its total resource use, biomass production rates, spatial distribution, and patterns of mortality and succession [29], [38]–[41]. For example, frequency distributions of trees follow characteristically similar relationships across forests in different regions experiencing different resource environments [29], [40], [41]. These frequency distributions follow a power-law over a large range of the data with a drop-off for the tallest trees [29], [40], [41]. This implies that the tallest trees can be used to infer the size structure of forests. Given the significance of maximum tree height our framework offers future extensions for understanding regional and global energy budgets, water and carbon cycles, temperature feedbacks, and ecosystem dynamics in response to changing environmental factors from the perspective of average physiology. It should be noted that our framework can be used to predict the variation of other plant features beyond maximum tree height such as the environmental variation of stomatal density.

Beyond its importance as a predictor of forest demographics, tree height has been shown to influence competition between individual trees for access to light [42]–[45]. However, the advantage of being taller comes with the added costs of growth and maintenance and this may set up a complicated evolutionary game between individuals [42], [43]. Maximum height has various correlations and related tradeoffs with other important plant traits [44], [45]. These include seed mass, overall growth rate, leaf mass per area, and wood density, each with environmental consequences ranging from soil resource use, to biomass production rates, to competitive dynamics within a community [44], [45]. Our framework provides insight into the environmental and physical limitations of these evolutionary dynamics.

In general, tree height is constrained by the interplay between many competing factors including resource limitations, internal metabolic constraints, overall growth rate, maturation processes, the hydrodynamic flow through vascular tubes of the branch network, its geometry and topology, and biomechanical and gravitational forces [22], [27], [28], [44]–[47], [47]–[49]. This complicated intersection of constraints is not unique to height but is a standard characteristic of most tree traits. Nevertheless, data on many properties of trees () can be encapsulated and summarized in phenomenological scaling laws which typically approximate a simple power law form:(1)where is tree mass, a normalization pre-factor, and the scaling exponent. Examples include tree heights (), respiration rates ( or ), overall growth rate (), the frequency distributions of individuals (), and trunk radii () [22], [25]–[27].

These scaling laws represent the average variation of a given evolved trait across many species. Because trees have simultaneously negotiated the limitations imposed by multiple physical constraints over their complicated evolutionary trajectory, these scaling laws are likely the manifestation of multiple constraints. For example the evolved canopy structure must be both mechanically stable and able to gather sufficient solar resources in order for the tree to survive and compete. Thus considerations of either or both of these limitations may anticipate an observed empirical scaling law. By focusing on empirical scaling laws these constraints, whether known or unknown, are then implicitly incorporated into our model without needing to specify which limitations – or combination of limitations – are the most important. Both hydraulic (e.g. [46], [48]) and mechanical (e.g. [27], [47]) limitations are argued to constrain maximum tree height and our model incorporates both of these via various scaling laws including the scaling of basal flow rates and the scaling of the canopy geometry. Beyond the inherent limitations of hydraulics or mechanics, ultimately and locally, maximum tree height is governed by the availability of resources. By connecting scaling laws to an interaction with the local environment we are considering the constraints of both resources and plant structure.

Model framework

We investigate the survival of an idealized tree with features determined primarily by its size. These features include the number of leaves, canopy shape and size, and the root mass, all of which interact with the environment via the tree's requirements for light and water (Fig. 3). Trees rely on their phloem and xylem for the internal distribution of nutrients and water. This circulation is a process of trees extracting moisture from the soil and making it available for evaporation, which drives the flow at the leaves. Accordingly, the rate of fluid flow through the vascular system has been a long-standing focus of environmental tree physiology [50], [51].

Our strategy is to compare flow rates that are constrained by resource supply with the flow rates that are required to sustain a tree of a given size in the absence of resource limitations. Both of these types of flow are governed by overall tree height according to scaling laws which relate various tree features to size. A basic assumption of our framework is that the essential tree traits required for building our predictive model scale with tree size according to approximate power laws (including isometric relationships). For many traits this is well supported by existing data. However, it should be noted that these power laws may break down for small trees where more complicated relationships hold (e.g. [27]) and some scaling exponents are known to have different values and confidence levels across different environments (e.g. [36]). These variations are beyond the scope of our efforts here. We focus on power laws because we are interested in the simplest construction of average behavior as a tool for predicting and understanding variation across species. Thus, we are testing the predictive power of the zeroth order approximation, which in this case are the widely used and studied power laws between body size and various plant traits. Future work should consider the higher order behavior of more complicated trait models.

Scaling relationships quantify how the total required flow rate of water in a tree, , changes with overall body size in order to support its basal metabolism [28], [39], [52]. We examine two principle limitations to the flow rate in trees: available water and energy (light and heat). Energy from the environment results in an evaporative flow rate of water through the tree, , which depends on both body size as well as on meteorological conditions, including air temperature, pressure, relative humidity, and solar radiation. This evaporative flow rate, which is the actual flow rate through a tree, must be met by a sufficient available flow rate of water from precipitation captured by the root mass, , which is also dependent on body size. In addition, must be sufficient to support basal metabolic needs encapsulated by . These constraints can be summarized as follows:(2)Thus, and set the boundaries of acceptable flow. Maximum tree height can then be predicted by finding the largest tree for which this relationship holds. In other words, our strategy searches for trees that use energy from the environment to meet their metabolic needs without exceeding their water resources. Fig. 2A summarizes our model, highlighting the factors involved in calculating , , and .

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 2. Schematics of the modeling framework.

(A) The relationships between the required flow rate, , the evaporative flow rate, , and the available flow rate, , and the factors which influence them. (B) Limitation Diagram. Red Curve: the flow rate of available water, which is a function of precipitation and size, as described in the text. Blue Curve: the required flow rate determined from allometric scaling, which is a function of size but independent of environmental conditions. Black Curve: the calculated evaporative flow rate, which is dependent on both size and meteorological conditions. The intersection of the black curve with either of the other two determines the maximum tree height.

https://doi.org/10.1371/journal.pone.0020551.g002

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. The size-based resource gathering capabilities of a tree.

The above-ground canopy is shown in green and the below-ground root mass in blue. The essential dimensions of the tree are indicated, where is the radius of the canopy, is the height of the canopy, and is the radius of the root mass. Each of these features scales with height, , where [41], and . The number of leaves scales as [28]. The scaling of the canopy features determines the collection of solar radiation and the heat exchange with the atmosphere, which can be used to solve for . The rate of moisture absorption, , is related to the scaling of the root system and incoming precipitation. Please see Supplement S1 for a more detailed treatment of these scaling relationships along with derivations for the associated tree physiology.

https://doi.org/10.1371/journal.pone.0020551.g003

Graphically, Eq. 2 implies that if we plot and as functions of tree height, , trees can only function in the region (the green-colored region of Fig. 2B). If we then plot a curve specific to a given environment, , we can determine which curve, or , is first intersected by at lower . The value of at this intersection specifies the height of the tallest possible tree. If a tree were to grow larger than this in the given environment, then its evaporative flow rate would exceed the availability of some resource. In water-rich environments lacking the appropriate incoming energy, intersects before it intersects , and this determines the maximum tree height. On the other hand, in water-limited environments with ample solar radiation, the reverse is true.

In order to explicitly calculate maximum tree height, we need to relate these various flow rates to tree height by invoking scaling relationships. Reference [28] provides a convenient way to relate height to several other dimensions of trees.

Basal metabolic requirements of a plant ()

The total basal volume flow rate of internal fluid is well approximated by(3)where is stem diameter, and are normalization constants, and and are scaling exponents [28], [29]. Empirically, best fits to data give , liter day cm with in cm, and liter day cm for in cm [39] (see Supplement S1). In order to convert the empirical relationship in equation 3, which relates to diameter, to a relationship concerning height we employ a calculation which relates various tree dimensions such as height and diameter. For large tree sizes it has been shown in [27] that which agrees with our analysis of (see Supplement S1). We rely on an analytic calculation to find because the model in [27] includes a small tree correction to the basic power law which is outside of the scope of our stated goal. When a direct empirical relationship between two features, such as and , is not known we typically employ an analytic calculation in order to avoid the propagation of error resulting from the combination of two or more empirical relationships. In some situations this is not possible because there are no known analytic derivations. Our overall framework, which is simply the connection of specific set of scaling relationships, does not depend critically on these analytic calculations. For future efforts one can employ our framework and replace any given empirical or analytic scaling relationship with alternative data or calculations. All that is actually required are the phenomenological scaling relationships themselves which are, or can be, constructed from data (all parameter symbols, definitions, and values can be found in Table S1).

Available flow rate due to precipitation ()

Given an incoming rate of precipitation, and ignoring hydrology (i.e., water due to runoff, pooling, or subterranean flow and storage), the moisture available to a plant is based on the capture area and capture efficiency of the root system. The capture area for precipitation is defined by the lateral extent of the root system, which can be determined from the geometric properties of the root architecture. From the data and scaling relationships given in [22], [27], [28], [53] the radial extent of roots is approximately given by(4)with (see Supplement S1 for detailed discussion). In our model, trees have access to the total volume of precipitation that falls on the area of flat ground directly above the root system, adjusted by the absorption efficiency of the roots. This can be expressed as(5)where (m year) is the rate of precipitation, and is the root absorption efficiency.

Evaporative flow rate ()

Trees act as passive solar pumps with the rate of water escaping due to evaporation equal to the internal flow rate. Hence, is governed by incoming energy. The basic physiological responses of tree canopies to local meteorology are well-established and are typically summarized using an energy budget [50], [51]. Although an energy budget formulation, which represents the overall conservation of energy, is conceptually simple, each individual energy flux requires a careful calculation based on the physics relevant to the appropriate tree characteristics, such as the density of stomata on a leaf and the geometry of the canopy. In Supplement S1 we provide details of these calculations which include considerations of both the tree size and environmental dependence of evaporation, radiation and conductance in the leaf and canopy microclimate. These are all governed by well-known physical laws, such as the Stefan-Boltzmann law for radiation, whose parameters have been measured or, in the few cases where they are not known, can be derived within our framework.

The basic energy budget requires that the total radiation absorption rate of a canopy, , is the sum of the rates of emitted thermal radiation and the sensible and evaporative heat losses:(6)Here, , , and are energy fluxes (W m): is the emitted thermal radiation, the sensible heat loss, and the latent heat loss with being the latent heat of vaporization for water and the evaporative molar flux (mol m s) [50], [51]. The coefficients , , are effective areas (m) over which each heat flux occurrs and are determined by considering how the canopy architecture affects the degree to which each flux is coupled to the atmosphere.

In terms of the molar mass, (kg mol), and density, (kg m), of water, the evaporative flow rate is related to by(7)From Eq. 6, we observe that the dependence of , and therefore , on tree height arises entirely from and the effective areas, , since , , and depend only on meteorological conditions. Thus, we can write(8)where represents the set of meteorological variables.

Each effective area for heat flux has a linear dependence on the total one-sided leaf area of the canopy, , where . The height dependence of can be determined by noting that(9)where (W m) is the incoming radiation per unit area (normal to the ground), is the absorption coefficient for the canopy, and is the projected area of the canopy. Both and depend on tree height via the shape of the canopy and the number and distribution of leaves within that canopy. For a given incoming radiation, for large trees, whereas, for smaller trees, a more complex, but derivable, relation holds (please note that capital the “” notation refers to absorbed radiation and should not be confused with lower-case “” which refers to root or canopy radii).

For the average tree whose features are encapsulated in the scaling relationships, these derivations have predictive power beyond determining maximum height. For example, our model predicts the specific form of the decrease in canopy albedo with increasing tree height in excellent agreement with data, as illustrated in Fig. 4 (please see Supplement S1 for a derivation). Albedo plays a critical role in many questions related to the earth system and our model framework provides a quantitative means for linking albedo to tree heights and thereby to local resources.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. The relationship between tree height and the total albedo for the canopy of a single tree.

The red points are data [67], and the blue curve is our generalized model for a tree using a soil reflection coefficient of and a deep canopy reflection coefficient of [51] (see Supplement S1). We have not included error bars here because ref. [67] does not provide them for every point.

https://doi.org/10.1371/journal.pone.0020551.g004

Because evaporation depends on many contributing meteorological variables (solar radiation, air temperature, relative humidity, and wind-speed) and on multiple tree traits (such as average leaf size and stomatal density) it is not possible to write a simple scaling relationship for the evaporative flow rate, . In determining we picked representative values for tree features that entered into the calculation and used the same values across all locations. (A detailed treatment of along with the parameter values used can be found in Supplement S1 in Table S1.)

Predicting maximum tree height and other traits

To determine maximum tree heights across the continental United States, we combined meteorological data sets (see Supplement S1) to calculate the functions and for the conditions at each location with determined from Eq. 3. As discussed above, our predictions for maximum tree height are found from the first intersection of with either or . We find that scales similarly to (Fig. S1) and that, in practice, the best predictions are achieved by searching for intersections of with once the root absorption efficiency, , has been established (see Supplement S1).

Because tree height spans nearly two orders of magnitude, we used the relative error, , to compare our predictions, , with observations, , of maximum tree height. As can be seen from the figures, our model gives good agreement with observed maximum tree heights, suggesting that it does indeed capture the essential features of environmental constraints and tree physiology. Fig. 1C shows a histogram of the relative error prior to taking absolute values () making it possible to determine over- and under-prediction. Error values are relatively narrowly distributed and the center of the distribution is close to zero. (Please see Supplement S1 and Fig. S2 for a discussion of the slight bimodal nature of this distribution.)

We tend to over-predict maximum tree height in wet environments where there are likely competitive factors limiting tree height. Under-prediction in our model generally occurs in arid environments where trees likely have developed specialized traits which deviate from the average values we used. However, with different, more realistic trait values, such as lower stomatal density in arid environments, we find that these trees obey Eq. 2. This is to be expected as different trait values are better suited to different environments. We can expand our framework by allowing traits to vary in order to optimize maximum height while still obeying Eq. 2. For example, holding all other tree parameters constant we can find the stomatal density which maximizes the upper bound on tree height in a given environment. We observe in Fig. 5 that the optimal stomatal density that we calculate decreases with increasing average annual temperature consistent with observations [54]. We also calculated the optimal leaf size in a similar fashion and found it to decrease with increasing temperature (not shown), which is also a trend suggested by observations [55]. This type of analysis, where the model is used as a point of departure for including sub-dominant effects, including the covariation of other traits, is an important area of investigation. In Supplement S1 we conduct a similar analysis to determine the optimal allometric scaling of two plant features which we initially took to be constant, the stomatal density and root absorption efficiency. We show that incorporating these additional scaling relationships into our model can reduce the error between predictions and observations (Fig. S5). Understanding the covariation and co-optimization of various plant scalings is an important area of ongoing [24], [36], [37] and future research.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 5. The change in the stomatal density as a function of environmental temperature.

The values presented are averaged over both sides of the leaf. The blue points are predictions from our model for the optimal stomatal density in each environment, holding all other traits at the average value. The red points are observations from ref. [54]. The observations fall within the range predicted by the model.

https://doi.org/10.1371/journal.pone.0020551.g005

Finally, we explore the effects of environmental shifts on maximum tree heights while holding plant traits constant. Applying the simplest case of a uniform change in mean annual temperature across the United States of C we can solve for the maximum height in that environment. We chose this value because C compares well with the conservative projections for temperature change over the next 100 years according to the frequently cited scenarios summarized by the Intergovernmental Panel on Climate Change (e.g. [56]). We find that for C the average maximum height across the continental U.S. decreases by 11 while for C the average maximum height increases by 13 (Fig. S3).

Scaling laws

For the empirical scaling laws used in this paper we have presented the error associated with scaling constants and exponents when the original reference provided this information.

Height and meteorological data

For observed maximum tree heights we used the United States Forest Service's Forest Inventory and Analysis (FIA) database, which records the height and location of individual trees [58], [59]. We are interested in predicting the largest tree in an area given local meteorology. The spatial variation of meteorology can be significant over relatively short distances. Thus, it is important to pair tree sites to meteorological stations which are geographically close to one another. This ensures that the predictions are capturing the conditions experienced by the observed trees. We paired trees with meteorological stations from the National Climatic Data Center (NCDC) [60] for purposes of using station or station interpolate data. Tree-meteorology pairs were separated by no more than 100 m of elevation and 4 km of radial distance. As a result of these stringent criteria we were only able to use a small subset of trees from the FIA database.

We considered all meteorology in terms of long-term annual averages. For precipitation we used the Parameter-elevation Regression on Independent Slopes Model (PRISM) [61], [62] 30-year average (1971–2000) sampled at the location of the meteorological stations. We constructed mean temperatures for individual stations using data from the NCDC [60]. We calculated relative humidity from the PRISM 30-year average [61], [62] for mean dewpoint temperature, minimum temperature and maximum temperature using a method described in ref. [50]. For wind speeds we used data from the National Centers for Environmental Prediction (NCEP) reanalysis [63], [64]. Solar radiation data was obtained from the NREL national grid [65].

Tree traits

Because of our focus on size and its relationship to survival in an environment we chose a single set of plant traits representative of a wide variety of tree species from different environments. This single set of traits was used across all environments to calculate . For each tree trait we examined the variation across many species, plant sizes, and environments and picked values that were representative of that variation. For several traits we checked that our values compared well to averages from the TRY database [66] which is a comprehensive collection of 65 trait databases and is representative of a large number of species and geographical regions. We picked traits that were appropriate for both angiosperms and gymnosperms, as our model does not distinguish between the two. We checked that the traits we picked gave rise to related properties, such as LAI, that were similar to observations for both angiosperms and gymnosperms. The values which we used can be found in Table S1 along with means from the TRY database.

Sensitivity analysis

It should be noted that many empirical scaling exponents come with associated error bounds and that these exponents can vary across different environments (e.g. [36]). In addition, we have relied on several analytic derivations to inform some of the scaled tree physiology. To deal with the potential variation of exponents we have carried out a basic sensitivity analysis where we perturbed individual exponents away from the value used for our predictions and examined the shift in the median relative error between observations and the new predictions (Fig. S4) (see Supplement S1 for details).