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The West, Brown, Enquist (WBE) model derives symmetrically self-similar branching to predict metabolic scaling from hydraulic conductance, K, (a metabolism proxy) and tree mass (or volume, V). The original prediction was K∝V0.75. We ask whether trees differ from WBE symmetry and if it matters for plant function and scaling. We measure tree branching and model how architecture influences K, V, mechanical stability, light interception and metabolic scaling.
We quantified branching architecture by measuring the path fraction, Pf: mean/maximum trunk-to-twig pathlength. WBE symmetry produces the maximum, Pf = 1.0. We explored tree morphospace using a probability-based numerical model constrained only by biomechanical principles.
Real tree Pf ranged from 0.930 (nearly symmetric) to 0.357 (very asymmetric). At each modeled tree size, a reduction in Pf led to: increased K; decreased V; increased mechanical stability; and decreased light absorption. When Pf was ontogenetically constant, strong asymmetry only slightly steepened metabolic scaling. The Pf ontogeny of real trees, however, was ‘U’ shaped, resulting in size-dependent metabolic scaling that exceeded 0.75 in small trees before falling below 0.65.
Architectural diversity appears to matter considerably for whole-tree hydraulics, mechanics, photosynthesis and potentially metabolic scaling. Optimal architectures likely exist that maximize carbon gain per structural investment.
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A large and growing body of research has focused on the coordination of hydraulic transport with the metabolism of photosynthesis and growth. While empirical research on this subject is quite extensive (e.g. Brodribb, 2009), a prominent component is metabolic scaling theory (MST) which stems from the original development by West, Brown & Enquist (WBE) (1997, 1999). The theory, as it applies to plants, centers on the premise that water transport is a co-limiting factor for photosynthesis. Because water transport is a largely physical process dependent in part upon transport network structure, its scaling can be predicted from relatively simple allometric models, leading to scaling predictions for all dependent metabolic processes.
The WBE model is fairly simple in its design. Plant branching structure is divided into external and internal components. The external structure follows symmetrical and self-similar branching (see Fig. 1a, rightmost tree) which allows the structure to be easily scaled. The external structure also conforms to biomechanical principles of area preservation and safety from gravitational buckling. The internal branching structure is the network of xylem conduits within the branches. The number and dimensions of xylem conduits are linked by simple rules to the external branch network (Savage et al., 2010; Sperry et al., 2012).
Central to MST are relationships described by power functions of the form y = axb where a is a scaling multiplier and b is a scaling exponent. Oftentimes, the focus is on the proportionality, y∝xb. The WBE model's prominent achievement is the analytical prediction in agreement with at least some empirical observations (Niklas & Enquist, 2001) that metabolic rate (B) scales with mass (M) to the 3/4 power (i.e. B∝M3/4; symbol definitions repeated in Table 1). This scaling prediction may be broken into two separate components that individually relate mass and water use to the easily measured dimension of trunk diameter, DT.
Table 1. Symbol definitions and modifiers from the main text in order of appearance
Basic symbols and definitions
total stem volume
‘volume exponent’ in
whole-tree sapflow rate
whole-tree hydraulic conductance
‘hydraulic exponent’ in
‘metabolic exponent’ in B∝K∝Vcq
path fraction = mean L↕ / maximum L↕
pathlength from trunk base to twig tip
branching junction furcation number
2 to 4
rank = number of supported twigs
minimum possible R for a given daughter in a given junction
maximum possible R for a given daughter in a given junction
probability of choosing a given daughter rank
exponent used to shift towards choosing A or Z
−5 to 5
maximum pathlength from branch base to twig tip
scaling multiplier (m1/3)
theoretical L↑ at which tree of given DT should buckle
Lcrit scaling multiplier (m1/3)
eventual safety factor from buckling
virtual length: distance beyond twig tip to theoretical origin (m)
stem segment length between junctions
scaling exponent for how diameter of main stem varies with height
scaling exponent for how supported mass varies with height
first positive root of Bessel function with input, ν
total absorbed photosynthetically active radiation (μmol s−1)
volume fraction = actual stem volume / volume of a column of equivalent height and basal diameter
The stem mass (and volume, V) is assumed to scale with . This ‘volume exponent’, c, is predicted to converge on 3/8, which is supported by theoretical and empirical considerations (McMahon & Kronauer, 1976; von Allmen et al., 2012). The rate of water use, Q, is assumed to scale with . The model predicts Q from whole-tree hydraulic conductance, K, which is calculated from internal vascular allometry. If the flow-induced pressure drop from soil to leaf is size invariant, then K ∝ Q. Because water loss and CO2 uptake utilize the same stomatal pathway, carbon assimilation should have a direct relationship to Q. If a constant fraction of photosynthate goes towards growth (a proxy for B) the result is . The product of the ‘hydraulic exponent’, q, and c gives the ‘metabolic exponent’, cq: B∝Q∝K∝Mcq. The WBE derivation of cq = 0.75 arises from the prediction that q converges on 2 for infinitely large trees. Thus, c = 3/8, q = 2 and cq = 0.75. Smaller values of q (0.68–1.91) and, hence, cq (0.25–0.70) are predicted for finite trees (Savage et al., 2010; Sperry et al., 2012).
Since its creation, revisions have been made to the WBE model, which have dealt with altering the branching structure within the confines of perfect symmetry (Price et al., 2007) and making the internal anatomy more realistic. The anatomical modifications have included more accurate scaling of xylem conduit number (Savage et al., 2010) and the addition of leaves, roots and nontransporting tissues (Sperry et al., 2012). These revisions have led to more accurate predictions (Price et al., 2007; von Allmen et al., 2012) but trees were still assumed to follow symmetrically self-similar branching. Real trees show average branching ratios (daughter/mother branch number, diameter and length) that can be similar to the constants predicted by WBE's symmetric self-similarity (Bentley et al., 2013). However, the distributions are quite broad, indicating a sizable fraction of asymmetric junctions. Even a few asymmetric junctions amongst major branches could significantly alter whole-tree symmetry.
We ask whether the branching architecture of real plants deviates substantially from the WBE structure. We then address the consequences of deviation with a model. We use the WBE model as a reference point and develop a novel numerical simulation method for building trees that represents the full range of tree morphospace from WBE symmetry to maximal asymmetry. Our numerical approach uses a minimum of deterministic branching rules and instead relies on probability distributions to build branch junctions and trees of varying symmetries. Our only major branching assumptions are that trees conform to the well-established patterns of area-preserving branching (Horn, 2000) and network-scale elastic similarity (McMahon & Kronauer, 1976). We use the improved internal anatomy of Sperry et al. (2012) but hold xylem parameters constant across simulated trees in order to isolate branching effects. We use the numerical model to investigate how deviations from WBE branching affect whole-tree hydraulic conductance, total stem volume, safety from gravitational buckling, and light interception. The model is also used to predict the influence of branching architecture on the scaling of tree hydraulic conductance (exponent q) and volume (exponent c) with trunk diameter, and hence how hydraulic conductance and its dependent processes scale with mass (exponent cq).
The path fraction index for tree form
We developed the ‘path fraction’, Pf, to quantify how much a particular branch network deviated from the WBE ideal. The path fraction is based on the pathlengths from twig tip to trunk base. We use the symbol, L↕, for this pathlength where the double arrow indicates that this length spans two extremes, twig tip to trunk base. In a WBE tree, all values of L↕ are the same. In our model, deviating from WBE by removing junction symmetry adds variation to L↕. We define the path fraction as
The bar in refers to the mean L↕ for the tree and the asterisk in (and other symbols that follow) indicates the maximum. The is an approximation of plant height so we will also use this symbol for height. The maximum possible Pf is 1, which occurs when (e.g. WBE trees; see Fig. 1a, rightmost tree). A high Pf corresponds to a round-shaped, spreading crown while a low Pf corresponds to a narrow crown with limited spread (Fig. 1). The minimum Pf is made by a structure with a central axis with twigs attached alternately. This structure minimizes and we refer to it as the ‘fishbone’ structure (e.g. Fig. 1a, leftmost tree). We use Pf as the independent branching structure variable against which we plot the functional attributes of tree hydraulic conductance, volume, mechanical stability and light interception.
Empirical path fractions
As a test of how much real plants deviate from the WBE structure, 40 Pf measurements were made of real branch systems. Specimens came from 15 different species and included both whole individuals and branches of open-grown trees and shrubs (species and sources in Table 2 and Supporting Information Notes S1). Species were chosen to represent a wide range of apparent architectures. Branches were obtained by a single cut just distal to a branch junction. Path fractions were obtained in two ways. For some (mostly the entire individuals) each segment between branching points was labeled and its length, diameter and mother segment were recorded. Twig-to-base paths were then reconstructed from these data to get all L↕ values. For the other specimens, L↕ values were measured directly by following stems from base to twig tips using a marked string with 10-cm precision. For this direct method, specimens were measured in spring so the measurements were made to tips that appeared to have been active the previous season.
Table 2. Empirical Pf measurements from trees and shrubs
Direct Pf measurements were time-consuming, limiting the size range to trees with trunk diameters, DT, < c. 5 cm. To estimate the Pf of larger trees to trunk diameters over 1 m, we used the crown area vs trunk diameter dataset of Olson et al. (2009) see Fig. S1) from angiosperm trees. From their published data (including all branches and trees in sheltered and salt-sprayed environments), we obtained an OLS regression to predict vertically projected crown areas from DT. We matched these predictions to 3D modeled trees with the same DT and within 5% of the same crown area. The Pf from these matching model trees were used to construct a Pf ontogeny.
Tree building model
Our tree building model was written in the R language (R Core Team, 2013) and is available from the senior author upon request. The model begins by sequentially defining junctions, starting with the trunk. At each junction, the mother branch (subscript m) splits into a number of daughters (subscript d). The number of daughters is f, the furcation number. Within each tree, we randomly chose a maximum furcation, f*, and then at each junction we chose f from 2 to f*. The f* was 2, 3 or 4, which covers the range for most botanical trees. Our f selection contrasts with the WBE model which uses a strictly constant f (n in their terminology). We assigned each branch an order or rank, R, equal to the number of twigs it ultimately supports (Katifori & Magnasco, 2012). Therefore, the starting point of each tree, the trunk, has Rm = the total number of twigs on the tree. This ranking system, illustrated in Fig. 2, simplifies tree building because: R is a finite integer; branch ranks change at each junction; and total rank is preserved across junctions. Each combination of mother rank, Rm, and f defines possible daughter ranks, Rd. Each daughter can only take on a certain number of different ranks because the sum of Rd must equal Rm. The first selected daughter rank, Rd,1, was always the smallest and was restricted to the range, A1 to Z1, where A1 = 1 and
where the floor brackets indicate the integer of the ratio. For subsequent daughter ranks, Rd,i, where 1 < i ≤ f, the Zi is given by
Equation 3 is just a variation on Eqn 2 where the numerator accounts for the fact that there is ‘less rank’ remaining to divide and the denominator indicates the ‘remaining rank’ is being divided among fewer undefined daughters. The intermediate values of Ai (if present) are different from the first and final Ai. For 1 < i < f, the Ai = Rd,i−1 as no daughter may be smaller than its predecessor. For the final daughter in the furcation, Rd,f, the Af = Zf such that Rd,f can only take on a single value that completes the mother rank.
The choice of Rd in each junction determines the symmetry of that junction. We controlled this choice by using a discrete probability distribution function to select each Rd,i at random from its respective Ai to Zi range. We defined this probability distribution with a power function because changing the exponent, u, allowed us to control the degrees of symmetry or asymmetry.
When u ≤ 0, the probability, P, of any Rd is given by
When u > 0, a slightly different equation is used,
For a given u > 0, Eqn 5 takes the probabilities from Eqn 4 with −u and mirrors them over the same A to Z range. For example, comparing u = 2 to u = −2 in a junction, when u = 2 is equal to when u = −2. When u < 0, asymmetrical junctions are favored while u > 0 favors symmetry. Using Eqns 4 and 5 with a u range of −5 to 5 created trees that populated the Pf range from maximum asymmetry (‘fishbone’ trees) to perfect symmetry (WBE trees). For a given tree, we fixed u at a single value. When u was varied within a tree to produce both strongly symmetric and asymmetric junctions, the generated trees were unrealistic (Fig. S2).
As an illustration of the daughter selection process, consider the circled junction in Fig. 2 (left tree). This tree has 10 twigs total and u = −2 was selected at random from −5 to 5. First, f* = 3 was selected from 2, 3 or 4 with equal probability. The f of the first junction (the trunk; with Rm = 10) was chosen between 2 and f* with equal probability. Choosing f = 3, the rank of the smallest daughter, Rd,1, was selected next. Because Rd,1 is the smallest and all daughters must add up to 10, Rd,1 must be between 1 (A1) and 3 (Z1), as given by Eqn 2. With negative u, Rd,1 = 1 will have the greatest probability ( from Eqn 4) and 3 will be very unlikely (). Suppose Rd,1 = 1 is chosen. The second daughter, Rd,2 is the next smallest so it may range from 1 to 4, as given by Eqn 3. Again, the minimum, 1, is most likely to be chosen. Here, Rd,2 = 2 was chosen. The final daughter has only one option, Rd,f = 7, resulting in a fairly asymmetrical junction. After creating this first junction, each daughter with Rd > 1 became a mother and junction selection continued, keeping f* = 3 and u = −2. The right tree in Fig. 2 shows how u = +2 can create much more symmetrical junctions.
After assigning all ranks, branch diameters and lengths were determined. Diameters were defined using constant twig diameters and area preservation (i.e. ). With R defined as the total number of supported twigs, each with constant cross-sectional area, R is proportional to the cross-sectional area of the branch. As such, diameter, D, is a function of R and twig diameter, DT:
This property is illustrated by the trees in Fig. 2 where diameters increase with R.
Length determination is more complicated but the guiding principle is that lengths must coordinate with diameters to achieve a constant safety factor from whole-tree elastic buckling from branch weight. Here, we define a new pathlength, , where the upward arrow indicates this length is from branch base (i.e. just above its lower junction) up to twig tip. This contrasts with the double arrow in L↕ which indicates trunk to twig path. The asterisk in signifies the maximum pathlength (i.e. to the most distant twig).
Empirical data indicate that once a trunk or branch reaches a modest D, its longest supported path, , tends to scale as (Niklas, 1994; von Allmen et al., 2012). The exponent of 2/3 is consistent with elastic similarity (i.e. constant deflection per length; McMahon & Kronauer, 1976). The critical height at elastic buckling, Lcrit, is also predicted to follow 2/3 scaling with D: Lcrit = bD2/3, where b can be explicitly calculated from tree form and wood properties (Greenhill, 1881). The shared 2/3 exponent means the safety factor from buckling () becomes constant at larger D. This ultimately constant safety factor, s, is equal to the ratio of the scaling multipliers: s = b/a. At smaller D, however, the by D scaling is steeper than 2/3. McMahon & Kronauer (1976) attribute this steeper exponent to a ‘virtual length’, lo. If the tree is represented as an elastically similar doubly tapered beam, then lo is the distance from the free end of the beam (i.e. the twig tip) to the point where the beam would taper to zero at its theoretical origin. McMahon & Kronauer (1976) show that by D scaling across all D can be fitted by an equation of the form:
As D increases, the lo term becomes comparatively negligible and the equation converges to (see Fig. S3).
Branch lengths were assigned from a single version of Eqn 7 (Eqn 8) that was applied across all trees regardless of their branching topology. The multiplier, a, was defined as a = b/s, where s = 4 and b was calculated from a WBE tree (b = 107.94 m1/3; see 'Mechanical stability of model trees' section below). The value of lo was derived from WBE trees (see Notes S2) and plugged into Eqn 7 to produce the by D equation for all modeled trees:
Equation 8 gives maximum length distal to each branch segment and from this, individual branch lengths (i.e. between junctions) were determined. At a given junction, the mother branch will have a certain and its daughters will have respective values. Because larger diameters support longer paths, it will be true that the daughter with the largest diameter, , will be part of the mother's longest path. Therefore, the segment length of the mother, lm, is
Twigs, which do not support daughters, have lengths equal to their :
The use of Eqn 9 can be illustrated by the left tree in Fig. 2. The trunk (Rm = 10) supports a maximum path of m (using model parameters in Eqn 8). Of its three daughters, only the largest daughter () lies along this path. This daughter supports a maximum path of m. Therefore, the length of the trunk segment must be the difference: lm = 0.10 m.
Hydraulic conductance of model trees
The hydraulic conductance, K, for each model tree was calculated from the internal network of xylem conduits. The internal anatomy is defined from the external anatomy following the recent WBE revision by Sperry et al. (2012). Briefly (see Notes S3 for details), hydraulic conductance of each stem segment is calculated from the diameter, number and length of functional xylem conduits (Savage et al., 2010; Sperry et al., 2012). Additional hydraulic resistances come from leaves, roots and conduit endwalls (Sperry et al., 2012). Segment conductances were combined using rules of network analysis to calculate K.
Sperry et al. (2012) used the external branching parameters of WBE to study the effects of variable internal anatomy. Here, we did much the opposite, using Sperry et al.'s default internal parameters while studying the consequences of branching pattern and Pf on hydraulic conductance and the hydraulic exponent, q.
Volume of model trees
Tree volumes were calculated to determine their sensitivity to Pf and, hence, the sensitivity of the volume exponent, c. Total stem volume, V, was the summed volume of all cylindrical branch segments. The volume of roots and leaves was not computed but assumed to be proportional to stem volume. If tissue density is invariant, then V becomes a proxy for stem (and plant) mass for purposes of metabolic scaling predictions.
Mechanical stability of model trees
The effect of branching structure on mechanical stability was assessed for all model trees by comparing estimated critical heights at elastic buckling (Lcrit) relative to estimated Lcrit of WBE trees (Lcrit,WBE). Typically, Lcrit is estimated by folding all branches up to make a column and assuming that the tree mechanically behaves as this column (Niklas, 1994). Furthermore, this column is assumed to have straight sides. To represent the full spectrum of more realistic trees, we used the alternative method of Jaouen et al. (2007), which identifies the ‘main stem’ (i.e. the thickest trunk-to-twig path) as the tallest mechanical structure which must support itself and all attached branches. The Jaouen et al. method accounts for the important effects of branching architecture on vertical mass distribution and Lcrit. The diameter, D, of the main stem may be described as a function of height, z, using
Likewise, the stem mass of all branches supported above z may be defined by
where Mtot is the total tree stem mass. The exponents n and m approximate the distributions of support capacity (D) and support requirement (M) in the main stem. For each tree, these exponents were calculated from Eqns 11 and 12 by standardized major axis (SMA) regression of logged data using the SMATR package for R (http://bio.mq.edu.au/ecology/SMATR/; Warton et al., 2006).
With some modifications to Jaouen et al.'s Eqn 1 (see Notes S4), we predicted Lcrit using
Values for the ratio of E (Young's elastic modulus; N m−2 and ρg (specific weight of supporting tissue; N m−3 for wood are approximately constant (Niklas, 1994). The cν (determined numerically in R) is the first positive root of the Bessel function of the first kind with parameter ν = (4n−1)/(m−4n+2) (Greenhill, 1881; Jaouen et al., 2007). The value of b in Eqn 8 corresponds to all the terms in front of in Eqn 13 where m, n, cν and Pf were from a WBE tree. When calculating Lcrit, two requirements were imposed. (1) Values of n and m are only meaningful when the data are well fitted by Eqns 11 and 12. We removed trees where fits had r2 < 0.95. (2) When ν < −1, the cν becomes somewhat erratic so these trees were also removed. Less than 7% of all modeled trees were removed for poor fits to Eqns 11-12 and only three trees in total were excluded for ν < −1.
Light interception of model trees
The importance of light interception is implied in the WBE model through ‘space-filling branching’ but it has not been quantified (Duursma et al., 2010). To estimate how Pf influenced light interception, we extended the model to three dimensions. For simplicity, we restricted 3D construction to trees where f* = 2 was chosen. Determining spatial structure required specification of branching angles and rotations with respect to connecting stem segments. Each branch segment was assigned an axis that runs along its length. ‘Branching angle’ shall refer to the angle a daughter axis makes away from its mother's axis. ‘Rotation’ refers to the rotation around its mother's axis. We adopted a set of maximally simple rules to set these angles and applied them equally across modeled trees. Thus, we emphasize the general effects of Pf on light interception and not secondary influences of branching angle variation.
To our knowledge, the only work that comes close to a general branching angle theory for plants is Murray's (1927) volume minimization equations (see also Zhi et al., 2001). However, these equations are inconsistent with area-preserving branching (two symmetric, area-preserving daughters are predicted to not diverge at all from their mother's axis). Nevertheless, Murray's (1927) Eqns 2-3 do produce realistic branch angle trends and so, despite their theoretical short-comings, we used them.
For rotation, daughters diverge from their mother's axis in opposite directions. Therefore, the daughters lie in the same plane. Accordingly, the mother also shares a plane with its sister branch. Each daughter plane was rotated 137.5∘ relative to its mother plane. The actual angle of rotation will depend on phyllotaxy and exactly which buds are released to form branches. However, our model is not an ontogenetic one and 137.5∘, the golden angle, is often observed and may minimize self-shading (Valladares & Brites, 2004).
As part of the 3D construction, we calculated crown area by projecting each tree from above and drawing a convex boundary linking the twig tips. Crown areas were used to estimate Pf from angiosperm crown scaling data of Olson et al. (2009; see ‘Empirical path fractions’, above). We also quantified tree shape as the aspect ratio (height/width). Height was actual height (instead of ), which was similar for all trees with equivalent twig numbers. Crown width was obtained from the diameter of a circle with equivalent area to the crown area.
The 3D trees were subjected to a light interception model using the turbid medium analogy (Campbell & Norman, 1998). Following Sinoquet et al. (2001), the three-dimensional space occupied by each tree was discretized into voxels (i.e. 3D pixels) of side length lvox. LAI of each voxel was calculated from the number of twig tips it contained and leaf area per twig (0.01m2). Interception by stems was ignored and we only modeled direct light with PPFD = 1500μmol PAR m−2s−1. To address the effect of source angle, we specified zenith angles every 3∘ from horizontal to directly overhead. For each zenith angle, we averaged light interception from eight azimuth angles. For each source angle, voxels were delineated to form columns parallel to the light source. As such, the LAI of each column of voxels was calculated. Absorbed PAR μmol s−1 is
(where Nc, the number of voxel columns; G, the ratio of projected and one-sided leaf area (Sinoquet et al., 2001)). Leaves were considered spherically arranged, making G = 1/2 for all columns and independent of source angle (derived from Monteith & Unsworth, 1990).
Using K, V and DT from the model, we tested how deviation from WBE branching affected the scaling exponents in: ; ; and K∝Vcq. We identified three scaling scenarios (S1, S2 and S3) for the relationship between Pf and tree size. Scenario S1 was a constant Pf with increasing tree size. Size-invariant Pf is perhaps most comparable to WBE scaling as WBE trees always have Pf = 1. We selected six target Pf values from 0.4 to 1.0. We then modeled 10 000 trees at each of seven twig counts from 26 to 212 twigs and isolated trees which had a Pf within 0.005 of each target. For trees with 25 twigs or fewer, Pf = 0.4 was not possible. The maximum twig number was limited by computation time.
Scenario S2 modeled the observed decrease in Pf with size from our Pf measurements. In this scenario, we fit a log function to our inter-specific Pf vs twig number data. We used this function to choose a target Pf at each modeled size up to 29 twigs (near the maximum in our data) and selected individuals that matched each Pf target ± 0.005.
In scenario S3, we used the Pf ontogeny estimated from Olson et al.'s (2009) angiosperm crown scaling data. The Olson et al. data covered a wider range of tree sizes. To accommodate this range, we built a limited set of 3D trees with up to 218 twigs (DT = 1024 mm). The subset of modeled trees that followed the crown scaling data showed a Pf-decreasing phase in small trees (as in our empirical measurements), followed by a Pf-increasing phase in larger trees (see Results). We defined the phase boundary at 211 twigs and modeled the scaling exponents separately for each Pf phase: 26−210 twigs (Pf decreasing) and 212−218 twigs (Pf increasing). For all scaling scenarios, we obtained q, c and cq from SMA regressions of logged data.
Measured, modeled and estimated path fractions
The Pf range is bound by WBE trees at the maximum (Pf = 1) and ‘fishbone’ trees at the minimum. Among modeled trees, a high Pf corresponded to a broad crown (aspect ratio near one) while low-Pf trees had narrower crowns (larger aspect ratio; Fig. 1a,b). Among our 40 Pf measurements from real plants, Pf ranged from 0.357 to 0.930. No specimen met either the WBE prediction or the ‘fishbone’ prediction. There was a significant trend for Pf to decrease with increasing size (Fig. 3, characters and solid regression line). While these data included both whole individuals (black) and branches (white), regressions fitted to each were not significantly different. Parallel to the observed decline in empirical Pf, the model predicts that as trees add twigs, the minimum possible Pf (the ‘fishbone’ structure) rapidly decreases from 1 before asymptoting around 0.25 (Fig. 3, shaded area). It makes sense that the potential to deviate from WBE becomes greater with more twigs. More twigs equals more and larger junctions and, therefore, more and greater opportunities to be asymmetrical.
Analysis of the Olson et al. (2009) data indicated a two-phase Pf trajectory (Fig. 3, dashed line). The first phase, in smaller trees (DT < c.6 cm), was a decline in Pf similar to what we measured. The second phase in larger trees showed a bottoming out of Pf followed by a gradual increase for DT > c.12 cm.
Pf and whole-tree hydraulic conductance
The model was run to produce 10 000 trees at each of nine different twig counts (24–212). However, to illustrate the functional consequences of Pf, we only show 1024-twig trees as a representative. Similar trends were evident at all modeled tree sizes. Deviation from WBE structure (i.e. lower Pf) tended to increase K (Fig. 4a) with the ‘fishbone’ structure having the greatest conductance and the WBE structure having the lowest. A more than two-fold increase was observed across the Pf range in the 1024 twig example with all trees having the same basal diameter and height. As Pf decreased, K increased because the average transport distance from trunk to twig decreased. Shorter average transport distances translated into higher average trunk-to-twig conductances. For each tree size, the K vs Pf data were fit with power functions. All fits were very good (r2 > 0.98). Some of the residual K variation was due to f* with linear regressions of residuals vs f* producing positive correlations with r2 = 0.30±0.09 (mean ± SD). Hence, larger f* tended to increase K at a given Pf. This is expected because greater f means branches become thicker (i.e. greater hydraulic conductivity) at a faster rate.
Pf and total stem volume
Reducing Pf caused V to decrease in a singularly linear fashion (Fig. 4b). Perfect linearity exists because of area-preservation and invariant twig diameters. As such, each L↕ represents a ‘tube’ of tissue with constant volume per length, as in the pipe model (Shinozaki et al., 1964). This relationship allowed us to define the volume fraction, Vf, as a corollary to Pf. The stem volume of each tree was standardized by the volume of a cylinder with equivalent height and basal diameter,
For modeled trees, Vf = Pf. The volume of a WBE tree is that of the reference column (i.e. Vf = 1). Other structures have lower Vf due to volumes less than the reference column (i.e. profiles more akin to a frustum).
Pf and mechanical stability
The Lcrit relative to the WBE tree was lowest near Pf = 1 (6.46% lower) and greatest near minimum Pf (30.97% greater; Fig. 4c). This Pf-dependent trend was due to the effects of m, n and total stem mass. The mass distribution exponent, m, was fairly constant across the Pf range: 2.97 ± 0.10 (mean ± SD; m = 1 corresponds to a straight column, m > 1 to a tapered column). Near Pf = 1, m was quite variable, which is reflected in Lcrit variability near Pf = 1 in Fig. 4c. Meanwhile, the main stem taper exponent, n, increased as Pf dropped (range = 0.93–1.23; n = 0 corresponds to a straight column). A larger n indicates stronger taper in the main stem and therefore less support tissue up high. This alone tends to reduce Lcrit. However, a smaller Pf indicates there is less total mass that requires support and therefore greater Lcrit.
Light absorption and Pf
Regarding light absorption, PARabs, we were limited to trees modeled in 3D (i.e. those with f* = 2). Trees with the same number of twigs also have the same total leaf area. Therefore, for a given number of twigs, PARabs variations are solely due to different leaf arrangements. At a given zenith angle, increased with Pf such that ‘fishbone’ trees absorbed the fewest photons and WBE trees absorbed among the most (Fig. 4d). This Pf effect increased as the light angle was shifted from horizontal side-illumination to overhead.
Scaling and Pf
We modeled three scaling scenarios: S1, Pf is constant through ontogeny of a particular species but can vary across species (Pf = 0.4−1.0; 26 to 212 twigs); S2, Pf decreases through ontogeny both within and across species, following the regression on our Pf data for small trees (DT < c.5 cm; 26 to 29 twigs; Fig. 3, solid line); and S3, Pf decreases in small trees (26–210 twigs) and reaches a nadir before gradually increasing in larger trees (212–218 twigs), as estimated from the Olson et al. (2009) data (Fig. 3, dashed line). In the three scenarios, the modeled data used for each scaling relationship (K by , V by and K by Vcq) were well fitted by power functions (r2 > 0.99).
The hydraulic exponent, q, was obtained from K by relationships. In S1, where Pf was constant with size, Pf = 1 predicted q = 1.80, which falls short of the original WBE prediction of q = 2 because of finite size effects and revisions to the internal anatomy (Savage et al., 2010; Sperry et al., 2012). As Pf decreased to 0.4, q increased to 1.85 (Fig. 5a); still shy of q = 2. In S2, Pf decreased with size, which caused K to increase at a faster rate than for constant Pf. Therefore, q steepened to 2.04: very near the WBE requirement. Similarly, in S3, as Pf decreased, hydraulic scaling steepened relative to constant Pf: q = 1.96. However, as Pf increased in larger trees, K increased more slowly and q decreased to 1.81.
Similar results existed for the volume exponent, c, in . In S1 (Pf constant through ontogeny), c was essentially unaffected by Pf: c = 0.364±0.001 (mean ± SD; Fig. 5b). All values were near but below the WBE prediction of c = 3/8 = 0.375. As shown by rearranging Eqn 15, and because Vf = Pf, an ontogenetically invariant Pf makes , meaning the scaling exponents among trees or species with different but constant Pf will be identical. Over the modeled size range, by DT is not a perfect power function because it has yet to converge on (Eqn 8). This fact, combined with a variable number of trees at each Pf-DT combination, made c < 0.375 and created some variation in c. In scenario S2 (Pf decreases through ontogeny), V increased at a slower rate relative to constant Pf, which lowered 1/c and increased c up to 0.41, exceeding the WBE prediction. When Pf decreased then increased (S3), the decrease produced a steeper c (0.389) followed by a flatter c (0.355) as Pf increased in larger trees.
The metabolic exponent, cq, in K∝Vcq, follows the q and c results. When Pf was constant through ontogeny, as in S1, cq showed a meager increase from 0.655 at Pf = 1 to 0.671 at Pf = 0.4 (Fig. 5c), well below the WBE prediction of cq = 0.75 due to the same finite-size effects as above. However, when Pf decreased throughout growth of smaller trees (S2), the larger q and c together exceeded the WBE prediction of 0.75: cq = 0.843. When Pf decreased and then increased with greater tree size (S3), cq initially exceeded 0.75 (0.760), before decreasing below all other values: cq = 0.642 (see Fig. S4).
In answer to our opening question, the results show that deviations from symmetrical WBE branching in real trees can be substantial and size dependent and these deviations have major effects on tree function and metabolic scaling. We used the path fraction, Pf, to quantify branching architecture in both real plants and modeled trees. We found that Pf in all of our real networks fell below the WBE ideal of Pf = 1. Furthermore, empirical Pf showed a biphasic ontogeny: first decreasing strongly with size before bottoming out at c. DT = 6−12 cm and gradually increasing thereafter. Our model predicted significant effects of deviating from the symmetrical self-similarity of the WBE model. When twig number was held constant (meaning constant height, leaf area and basal diameter), deviating from WBE led to greater whole-tree hydraulic conductance (K), lower stem volume (V), greater critical buckling height (Lcrit), and reduced total photon absorption (PARabs). When we ‘grew’ trees to different sizes we found that if Pf was held constant, deviations from WBE branching caused only a minor increase in the metabolic exponent, cq, owing to shifts in q. All cq values were below the original WBE prediction of 0.75. This was true even for WBE-branching trees because of finite size effects (Savage et al., 2010) and hydraulic architecture modifications (Sperry et al., 2012). If we assumed that Pf declined to a minimum before increasing with size, as observed interspecifically, the cq was size dependent. For small trees with decreasing Pf, cq could increase beyond 0.75 due to large increases in both c and q. But for larger trees with gradually increasing Pf, cq was much lower, falling below 0.65.
The ‘U’ shaped Pf trajectory estimated for real trees makes intuitive sense. Young trees may place a premium on height growth, which would be favored by Pf-decreasing crowns that become elongated and relatively narrow (Fig. 1b; Charles-Dominique et al., 2012). High hydraulic conductance per tissue volume, and greater mechanical stability (or greater height per trunk diameter) of low-Pf crowns may also maximize height growth. As the tree reaches or exceeds the height of surrounding vegetation, broader high-Pf crowns would capitalize on greater light availability and fill canopy gaps.
For a given DT and height, decreasing Pf was a strong predictor of increasing K. This result was somewhat surprising as Pf represents an entire branching structure with just mean and maximum pathlengths. However, the hydraulic conductance of a nontapering tube is inversely related to its length so it follows that K should increase as pathlengths are shortened (Fig. 4a). Indeed, individual trunk-to-twig hydraulic conductances within a tree were always negatively correlated with path length (not shown). In the model, the length-dependence of K is reduced but cannot be eliminated or reversed by observed xylem conduit taper or leaf resistance (Sperry et al., 2012). Data on path conductance and actual path length are limited, but support the prediction of greater conductance for shorter trunk-to-leaf paths (Sperry & Pockman, 1993). Whole-path conductance to branches lower in the canopy can be equivalent to (Hubbard et al., 2002; Yoshimura, 2011) or even lower than (Kupper et al., 2005; Sellin & Kupper, 2005) branches higher up, but the pathlengths were not measured in these studies. However, it is possible that shorter paths could develop lower conductances if shading caused senescence or growth of narrower twigs (Protz et al., 2000). Variation in twig properties within a canopy was not modeled, but could obscure the pathlength effect.
At the whole-tree level, the xylem architecture component of the model is known to yield realistic ranges of water use and K across different functional tree types (von Allmen et al., 2012; Sperry et al., 2012). Rigorous tests of the additional effects of branching await information on the ranges of Pf across major tree types. The Pf is a novel metric and there are no data on it outside of this paper. Although it is difficult to measure on large trees, in principle the model can be used to estimate it from the allometries of crown area and height vs trunk diameter. In real trees, the effects of variable branching structure are superimposed on effects of variable xylem anatomy. A virtue of the model is the ability to separate out the hydraulic contributions of these two networks.
The model showed that total stem volume decreased as Pf decreased (Fig. 4b). Trees with shorter transport distances on average require less construction tissue, even for the same height and basal diameter. Furthermore, V vs Pf was a perfect linear relationship, resulting in the volume fraction, Vf, being equal to the path fraction, Pf. The Vf is potentially much easier to estimate than Pf, which would facilitate its measurement in trees.
As Pf was decreased, the model also predicted that critical heights, Lcrit, increased for a given DT (Fig. 4c). The greater mechanical stability of conical low-Pf trees is an intuitive result because they carry more of their mass closer to the ground than round-crowned high-Pf trees. In the scenario we modeled, all trees of a given DT were the same height. Therefore, the increase in Lcrit resulted in greater safety from buckling in low-Pf trees. Alternatively, if trees grow towards the same safety from buckling, low-Pf trees should grow taller for a given DT than high-Pf trees.
The latter prediction appears to be supported by the available data. Among temperate trees, evergreens (mostly conifers) have been shown to grow taller with diameter than deciduous (mostly angiosperm) trees (Ducey, 2012). Although phenology was stressed in that study, our model suggests an alternative explanation: a tendency for large conifers to have a lower Pf may allow them to grow taller than angiosperms for the same trunk diameter. Lower Pf for large conifers is suggested by their tendencies to be taller (Ducey, 2012) and to have narrower crowns (see Fig. S1; Krajicek et al., 1961; Vezina, 1962; Leech, 1984; Farr et al., 1989) than similarly large-trunked angiosperms. Low Pf in large conifers would favor height growth per basal diameter by: increasing Lcrit; decreasing volume investment; and increasing tree hydraulic conductance.
PARabs was also influenced by Pf (Fig. 4d). In WBE trees, symmetric branching means there is no distinct main stem and branches can spread large distances in all directions. As Pf is lowered, a distinct main stem starts to develop with branches extending from this stem (see Fig. 1a). This configuration limits the horizontal spread of branches, leading to more self-shading of leaves and less light absorption (Pearcy et al., 2004a). Changing the light angle from vertical to horizontal reduced this disadvantage of low-Pf trees, but did not eliminate it. Lowering Pf should always limit the lateral spread of leaves, so using a different light model or alternative branching angles should not impact our general prediction.
Our results suggest how the local environment may select for optimal branching architecture. For three out of the four modeled tree properties (hydraulic conductance, volume, mechanical stability and light interception), low-Pf trees are at a competitive advantage as they can transport water more easily despite a smaller investment in tissue and have greater mechanical stability. However, these advantages come at the expense of reduced light absorption. Hypothetically, the diverse spectrum of tree forms in nature could result from optimizing this tradeoff for a diverse set of requirements, depending on life history and habitat (Horn, 1971). In general, selection for a given branching architecture will depend on the relative advantages of transporting water, growing fast, growing tall and gathering light. As already discussed, the optimal Pf of an angiosperm canopy tree may change through ontogeny, with decreasing Pf favoring early height growth followed by increasing Pf to favor canopy gap-filling (Horn, 1971). Alternatively, short shade-tolerant species adapted to the high humidity and low light of the understory are expected to always have a high Pf. Such species lack a prolonged height growth phase and need to avoid self-shading (Pearcy et al., 2004b). The associated low hydraulic conductance would not be a liability for short stature and low evaporative demand. Conversely, high-Pf shrubs or treelets would also be expected in open habitats where competition for light is absent and height growth is less advantageous.
While it was beyond the scope of this study to fully quantify the tradeoffs of different architectures, the concept of a Pf optimum can be illustrated by normalizing PARabs by V. Fig. 6 shows broad peaks at midrange Pf for all light angles. Moving the light source from vertical to horizontal sharpened and elevated the peak and shifted it to lower Pf. Increasing the tree size had comparatively little effect on peak shape or position (not shown). As PARabs is closely tied to photosynthesis and V to mass, the results are suggestive of peaks in carbon gain per carbon spent. Photosynthesis will also depend on water supply to the leaves and its influence on stomatal conductance. As such, the higher K associated with lower Pf (Fig. 4a) would tend to further benefit the midrange-Pf trees relative to high-Pf trees.
Systematic changes in branching architecture with size, either through ontogeny or across species, have potentially major effects on metabolic scaling. With size-invariant Pf, changing Pf had fairly small effects on the hydraulic and metabolic exponents (q and cq). Much larger effects have been seen by changing the internal structure such as xylem conduit taper and sapwood area scaling (Sperry et al., 2012). This result offers some support that the scaling of the WBE tree can be reasonably representative of non-WBE branching structures. This may explain why retaining the WBE structure resulted in generally good fits to sapflow data (von Allmen et al., 2012). The result also shows agreement with Bentley et al. (2013) that junction asymmetry is not a predictor of whole-tree scaling. However, the Pf appears to change systematically with tree ontogeny, making scaling exponents size-dependent and allowing cq to reach or exceed the original WBE prediction of cq = 3/4 in small trees. Within the constraints of WBE architecture, the only other identified mechanism of cq ≥ 0.75 in a finite individual is ontogenetically increasing the root-to-shoot hydraulic conductance ratio (Sperry et al., 2012).
The model quantifies basic trade-offs between branching structure and major aspects of tree function. Narrow, elongated crowns are predicted to maximize vascular supply and mechanical stability, and minimize tissue investment. Broad, round crowns maximize light interception. No single shape is likely to be optimal across all habitats and tree sizes, and shape appears to shift though ontogeny. The model provides a framework for ultimately predicting optimal architectures. Although differences in architecture exist across at least some functional tree types (e.g. angiosperm vs conifer), such variation needs to be expressed in terms of path fractions or the equivalent for a functional analysis. Our branching structure analysis adds another layer of complexity to the evolving theory of metabolic scaling in trees. The central, elegant predictions of the original WBE model for 3/4 scaling become fascinatingly complex when the variable structures of real plants are considered.
Mathematical interpretations were improved by discussions with Fred Adler. Peter Reich provided helpful input at various stages of the project. Jake Olsen and Erica von Allmen assisted with empirical measurements from Utah. We thank Will Driscoll for helping collect the ponderosa pine data. D.D.S. and J.S.S. were supported by NSF IBN-0743148. Funding from ATB Award 0742800 helped develop the initial ideas for this work.