## Introduction

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* = *ax*^{b} where *a* is a scaling multiplier and *b* is a scaling exponent. Oftentimes, the focus is on the proportionality, *y*∝*x*^{b}. 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*∝*M*^{3/4}; 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, *D*_{T}.

Basic symbols and definitions | Input | |
---|---|---|

B | metabolic rate | |

M | mass | |

D | stem diameter | |

V | total stem volume | |

c | ‘volume exponent’ in | |

Q | whole-tree sapflow rate | |

K | whole-tree hydraulic conductance | |

q | ‘hydraulic exponent’ in | |

cq | ‘metabolic exponent’ in B∝K∝V^{cq} | |

P _{ f } | path fraction = mean L_{↕} / maximum L_{↕} | |

L _{↕} | pathlength from trunk base to twig tip | |

f | branching junction furcation number | 2 to 4 |

R | rank = number of supported twigs | |

A | minimum possible R for a given daughter in a given junction | |

Z | maximum possible R for a given daughter in a given junction | |

probability of choosing a given daughter rank | ||

u | exponent used to shift towards choosing A or Z | −5 to 5 |

maximum pathlength from branch base to twig tip | ||

a | scaling multiplier (m^{1/3}) | 26.99 |

L _{ crit } | theoretical L_{↑} at which tree of given D_{T} should buckle | |

b | L_{crit} scaling multiplier (m^{1/3}) | 107.94 |

s | eventual safety factor from buckling | 4 |

l _{ o } | virtual length: distance beyond twig tip to theoretical origin (m) | 0.34 |

l | stem segment length between junctions | |

n | scaling exponent for how diameter of main stem varies with height | |

m | scaling exponent for how supported mass varies with height | |

c _{ ν } | first positive root of Bessel function with input, ν | |

PAR_{abs} | total absorbed photosynthetically active radiation (μmol s^{−1}) | |

V _{ f } | volume fraction = actual stem volume / volume of a column of equivalent height and basal diameter | |

Subscript modifiers | ||

T | trunk | |

m | mother | |

d | daughter | |

t | twig | |

Superscript modifier | ||

* | maximum |

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 CO_{2} 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*∝*M*^{cq}. 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*).