## Introduction

The crown architecture of plants is highly diverse, but it remains unclear how this diversity affects light interception and growth across species. Leaf display depends on a multitude of morphological traits (Hallé*et al.*, 1978; Barthélémy & Caraglio, 2007; Valladares & Niinemets, 2007), but it is difficult to generalize how these traits influence the light interception efficiency of individual plants. Moreover, different crown traits may result in similar light interception efficiencies (light interception per unit leaf area), suggesting functional equivalence of different architectural layouts (Valladares *et al.*, 2002). In this paper, we develop a coherent theoretical framework that allows the diversity in crown architectures seen among individual plants to be compared, in terms of their light interception efficiency. As plant productivity over long time-scales is approximately proportional to intercepted light (Monteith, 1977; Cannell *et al.*, 1987), it is hoped that such a framework can provide a basis for understanding differences in productivity between plants and vegetation types.

Although plants vary in a myriad of qualitative architectural properties, models predicting light interception typically focus on a small number of quantitative features of the canopy, such as total leaf area, leaf angle distribution, and leaf dispersion (clumped, random, or regular) (Campbell & Norman, 2000), as these can be reliably quantified for different species and vegetation types. The well-known Lambert–Beer model estimates light interception by horizontally homogeneous canopies, assuming that leaves are randomly distributed in space (Monsi & Saeki, 1953, 2005):

*Q*_{int}, intercepted photosynthetic photon flux density (PPFD); *Q*_{0}, incident PPFD above the canopy; *L*, the leaf area index; *k*, an extinction coefficient.) However, random leaf spacing is clearly a poor assumption for most real canopies, where foliage is clumped at shoot and whole-plant levels. As a result, model errors can be very large (Baldocchi *et al.*, 1985; Whitehead *et al.*, 1990; Cescatti, 1998). For this reason, a leaf dispersion parameter is frequently introduced in Eqn 1 (Nilson, 1971; Ross, 1981). The leaf dispersion parameter is typically estimated from inversion of PPFD transmission measurements in canopies (Nilson, 1971; Cescatti & Zorer, 2003), and is rarely related to direct measurements of canopy structure. A notable exception is a shoot clumping parameter developed for conifers (Oker-Blom & Smolander, 1988; Stenberg, 1996), but it is not clear how this parameter could be applied to other plant architectures. To this end, Sinoquet *et al.* (2007) developed a method based on the spatial variance of foliage in tree crowns, but this method seems to be difficult to apply to field measurements. The lack of a simple operational method to account for grouping of foliage is thus limiting our ability to link canopy light interception to plant and canopy structure.

A few attempts have been made to provide simplified models that account for grouping of foliage at the whole-plant scale (Jackson & Palmer, 1979; Kucharik *et al.*, 1999; Chen *et al.*, 2008; Ni-Meister *et al.*, 2010). However, the resulting models are often complex, with many species-specific parameters that are time-consuming to obtain. Furthermore, available models do not provide estimates of light interception by individual plants, limiting their usage to stand-scale applications. Estimates of light interception on an individual plant level are needed in individual-based models of vegetation dynamics (e.g. Moorcroft *et al.*, 2001; Falster *et al.*, 2011). A different field of study has avoided simplification altogether by developing highly detailed three-dimensional plant models with spatially explicit representation of leaves and stems. These studies have provided valuable insights into the details of plant architecture and how it affects plant performance in specific environments (see reviews by Valladares & Pearcy, 1999; Pearcy *et al.*, 2005; Vos *et al.*, 2010). However, detailed architecture studies are yet to discover which aspects of canopy structure most influence whole-plant light interception, in part because of the limited sample sizes that necessarily result when using such methods.

Generally speaking, plant crowns are defined by the number, size, shape, three-dimensional distribution and orientation of their leaves. Together, these variables determine the size of the crown, the arrangement of leaves inside the crown, and the average leaf overlap (‘self-shading’) when viewed from a given direction. We define light interception efficiency as the ratio of displayed (i.e. exposed) to total leaf area, averaged over the entire sky hemisphere () (Farque *et al.*, 2001; Delagrange *et al.*, 2006) (see Table 1 for a list of symbols). relates directly to the amount of diffuse radiation intercepted by the plant, which can play an important role in determining total carbon uptake (Ackerly & Bazzaz, 1995; Roderick *et al.*, 2001). Direct light interception also scales with the sunlit leaf area fraction (Campbell & Norman, 2000), which is probably correlated with . Simple indices of self-shading – such as have also been shown to predict total carbon uptake. For example, comparing branches from 38 perennial species, Falster & Westoby (2003) found that > 90% of variation in whole-branch CO_{2} assimilation rate expressed per unit leaf area (excluding differences in leaf photosynthetic capacity) was accounted for by an index of self-shading. has also been used in other applications to rank light interception efficiency of whole plants (Delagrange *et al.*, 2006; Sinoquet *et al.*, 2007) and shoots (Niinemets *et al.*, 2005), suggesting that it provides a reliable indicator of plant performance.

Symbol | Definition | Units |
---|---|---|

L | Leaf area index | m^{2} m^{−2} |

A_{C} | Crown surface area – total surface area of 3D convex hull wrapped around the leaf cloud | m^{2} |

A_{L} | Total plant leaf area | m^{2} |

A_{D,Ω} | Displayed leaf area from angle Ω | m^{2} |

H_{Ω} | Crown silhouette area from angle Ω | m^{2} |

P_{Ω} | Crown porosity from angle Ω | – |

Displayed leaf area averaged over all angles | m^{2} | |

a_{L} | Mean leaf area of individual leaves | m^{2} |

N | Total plant leaf number | – |

K | Leaf projection coefficient averaged over all viewing angles | m^{2} m^{−2} |

k | Extinction coefficient for a homogenous canopy | m^{2} m^{−2} |

k_{Ω} | Leaf projection coefficient from angle Ω | m^{2} m^{−2} |

Silhouette to total area ratio, averaged over all viewing angles | m^{2} m^{−2} | |

β | Leaf dispersion parameter | – |

ε | Empirical coefficient | – |

φ | Empirical coefficient | – |

Ω | Viewing angle (elevation, azimuth pair) | – |

f_{Ω} | Weighting function for A_{D,Ω} | – |

α | Solar elevation | ° |

O_{5} | Observed average distance to five nearest leaves | m |

E_{5} | Expected average distance to five nearest leaves for a random distribution | m |

As possible predictors of , we propose two simple whole-plant variables: crown density and leaf dispersion. These variables were selected based on a statistical model predicting shading within the crown, based on similar models independently developed by Sinoquet *et al.* (2007) and Duursma & Mäkelä (2007), but with a new leaf dispersion parameter. The model is derived (see the Materials and Methods section) by viewing the plant first from one direction, and estimating the leaf area displayed in that direction from the silhouette of the crown, the number of leaves, the mean leaf area, and the mean leaf angle. After making simplifying assumptions, this leads to an approximation for the average leaf area displayed in all directions:

(*A*_{L}/*A*_{C}, the crown density (ratio of plant leaf area, *A*_{L}, to crown surface area, *A*_{C}); β, a leaf dispersion parameter; ε and φ, empirical parameters.) The ‘extinction coefficient’*K* is constant because we integrate over the entire hemisphere (Stenberg, 2006) (see the Materials and Methods section). The crown surface area is defined as the total surface area of a crown, that is, the area of a sheet wrapped around the crown. It is calculated as the surface area of the three-dimensional shape constructed by joining all outlying points of the plant crown, so that the shape is convex (i.e. there are no indentations in the three-dimensional shape). The remarkable aspect of the approximation (Eqn 2) is that only two plant variables, crown density (*A*_{L}/*A*_{C}) and leaf dispersion (β), are needed to estimate , in addition to the two (constant) parameters. In this paper we test the hypothesis that, across plants of diverse architecture, size, and growth environments, variation in can be explained by crown density and leaf dispersion. To do so, we used a database of 1831 virtual plants, reconstructed from precise digitization of the position and orientation of leaves and stems of real plants, to estimate (cf. Farque *et al.*, 2001). We then compared these empirical estimates to those given by the simplified model. Finally, we tested the hypothesis that declines with increasing plant size for a given species (Farque *et al.*, 2001; Niinemets *et al.*, 2005; Delagrange *et al.*, 2006; Sinoquet *et al.*, 2007), and whether this decline is related to changes in leaf dispersion or crown density.