# A General Leaf Area Geometric Formula Exists for Plants—Evidence from the Simplified Gielis Equation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{b}, where b is a constant that is larger than one [2]. Thus, specific leaf weight is proportional to leaf area to the power b–1. That is, the photosynthetic rate increases with increasing LA. Since LA is closely associated with plant photosynthesis, it is important to calculate its value in practice. There are two types of common tools for computing the blade area of plants: non-handheld and handheld leaf area scanners [3]. A non-handheld leaf area scanner can calculate LA on the basis of the computational formula for polygonal area, but it is a destructive procedure. If one needs to observe continuous growth dynamics, a non-handheld leaf area scanner is inappropriate. Although a handheld leaf area scanner enables nondestructive LA measurements, its accuracy seriously depends on the background color setting and the flatness of the leaf surface. In the field, it is usually inconvenient to set the background color, and actual plant leaves are not naturally flat. If there were an LA geometric formula that requires measuring only some simple morphological parameters such as leaf length and leaf width, it would be convenient for computing leaf area in field experiments. Dornbusch et al. [4] proposed a six-parameter leaf shape model that can delineate leaf shapes of wheat, barley, and maize. Gielis [5,6] provided a superformula that can describe a variety of object shapes such as cells, leaves, flowers, and tree-ring cross sections. Shi et al. [7] and Lin et al. [8] confirmed the validity of a simplified Gielis equation (SGE) with two model parameters to describe the leaf shapes of 46 bamboo species.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Models

_{w}linked to leaf length and leaf width. However, a numerical solution for the parameter can be obtained. We can set a group of candidates of φ in small increments (e.g., 10

^{−6}) from 0 to π/2. Then, we calculate the approximate value of leaf width based on Equations (4) and (5) by minimizing the absolute value of the difference between the actual measured leaf width and the predicted values by Equation (5). Then, we can obtain the angle associated with the minimal absolute value of the differences, which is the numerical solution of φ

_{w}. As a consequence, the parameter n can be obtained according to Equation (4), and then l

_{w}can also be obtained according to Equation (2). The LA formula can be expressed as:

#### 2.3. Statistical Analyses

## 3. Results

## 4. Discussion

#### 4.1. Influence of the Prediction Error in Leaf Length on Computing the Leaf Area

#### 4.2. Beyond the Power Law between Leaf Length and Leaf Area

_{w}can be treated as a function of the ratio of leaf width to leaf length (i.e., the W/L ratio). Then, n is actually a function of W/L, on the basis of Equation (4). In this case, whether there is a ‘power of 2′ law between LA and leaf length might mainly rely on the shape of the distribution of n or that of the ratio of W to L. If the coefficient of variation of the ratios (CV = σ/μ, where μ and σ represent the mean and the standard deviation of the leaf width/length ratios, respectively) is very low, which means that the distribution curve of the leaf width/length ratios is narrow, we would expect that the estimate of the scaling exponent between leaf length and leaf area approximates 2 (Figure 8a,b). In this case, n values for different leaves vary in a small range. We also checked whether the extent of skewness of the distribution of the leaf width/length ratios can affect the estimate of slope. Skewness (S

_{k}) of the leaf width/length ratios was measured by the following equation:

#### 4.3. Comparison between a Non-Parametric Model and the Gielis Equation

_{SGE}replicates and RMSE

_{GAM}replicates included zero (Figure 9g–i), which demonstrates that there was no significant difference between the SGE and GAM [25,26]. That is, the SGE is a parametric model with general applicability for describing the leaf shape of many plants. If there was a small variation in leaf length (i.e., a small MPEL), the difference in RMSE between the two methods would be smaller (Figure 9d,f); if the MPEL was large, the RMSE predicted by the SGE would be slightly higher than that predicted by the GAM (Figure 9e). It is necessary to point out that the RMSE values using the GAM to fit the training set were all approximately zero, but the RMSE values using the above GAM to fit the test set were substantially larger than the former ones (Figure 9a versus Figure 9d, Figure 9b versus Figure 9e, Figure 9c versus Figure 9f). The non-parametric fitting method might have led to over-fitting to a certain extent (see the RMSE using the GAM in Figure 9a–c).

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Jurik, T.W. Temporal and spatial patterns of specific leaf weight in successional northern hardwood trees species. Am. J. Bot.
**1986**, 73, 1083–1092. [Google Scholar] [CrossRef] - Milla, R.; Reich, P.B. The scaling of leaf area and mass: The cost of light interception increases with leaf size. Proc. R. Soc. Biol. Sci.
**2007**, 274, 2109–2114. [Google Scholar] [CrossRef] [PubMed] - Schrader, J.; Pillar, G. Leaf-IT: An Android application form measuring leaf area. Ecol. Evol.
**2017**, 7, 9731–9738. [Google Scholar] [CrossRef] [PubMed] - Dornbusch, T.; Watt, J.; Baccar, R.; Fournier, C.; Andrieu, B. A comparative analysis of leaf shape of wheat, barley and maize using an empirical shape model. Ann. Bot.
**2011**, 107, 865–873. [Google Scholar] [CrossRef] [PubMed] - Gielis, J. Inventing the Circle: The Geometry of Nature; Geniaal Press: Antwerpen, Belgium, 2003. [Google Scholar]
- Gielis, J. The Geometrical Beauty of Plants; Atlantis Press: Paris, France, 2007. [Google Scholar]
- Shi, P.; Xu, Q.; Sandhu, H.S.; Gielis, J.; Ding, Y.; Li, H.; Dong, X. Comparison of dwarf bamboos (Indocalamus sp.) leaf parameters to determine relationship between spatial density of plants and total leaf area per plant. Ecol. Evol.
**2015**, 5, 4578–4589. [Google Scholar] [CrossRef] [PubMed] - Lin, S.; Zhang, L.; Reddy, G.V.P.; Hui, C.; Gielis, J.; Ding, Y.; Shi, P. A geometrical model for testing bilateral symmetry of bamboo leaf with a simplified Gielis equation. Ecol. Evol.
**2016**, 6, 6798–6806. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nicotra, A.B.; Leigh, A.; Boyce, C.K.; Jones, C.S.; Niklas, K.J.; Royer, D.L.; Tsukaya, H. The evolution and functional significance of leaf shape in the angiosperms. Funct. Plant Biol.
**2011**, 38, 535–552. [Google Scholar] [CrossRef] - Jones, C.S.; Bakker, F.T.; Schlichting, C.D.; Nicotra, A.B. Leaf shape evolution in the South African genus Pelargonium L’ Hér. (Geraniaceae). Evolution
**2008**, 63, 479–497. [Google Scholar] [CrossRef] [PubMed] - Runions, A.; Fuhrer, M.; Lane, B.; Federl, P.; Rolland-Lagan, A.-G.; Prusinkiewicz, P. Modeling and visualization of leaf venation patterns. ACM Trans. Gr.
**2005**, 24, 702–711. [Google Scholar] [CrossRef] - Runions, A.; Tsiantis, M.; Prusinkiewicz, P. A common developmental program can produce diverse leaf shapes. New Phytol.
**2017**, 216, 401–418. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Brodribb, T.J.; Feild, T.S.; Jordan, G.J. Leaf maximum photosynthetic rate and venation are linked by hydraulics. Plant Physiol.
**2007**, 144, 1890–1898. [Google Scholar] [CrossRef] [PubMed] - Price, C.A.; Symonova, O.; Mileyko, Y.; Hilley, T.; Weitz, J.S. Leaf extraction and analysis framework graphical user interface: Segmenting and analyzing the structure of leaf veins and areoles. Plant Physiol.
**2011**, 155, 236–245. [Google Scholar] [CrossRef] [PubMed] - Hastie, T.; Tibshirani, R. Generalized Additive Models; Chapman and Hall: London, UK, 1990. [Google Scholar]
- Shi, P.; Zheng, X.; Ratkowsky, D.A.; Li, Y.; Wang, P.; Cheng, L. A simple method for measuring the bilateral symmetry of leaves. Symmetry
**2018**, 10, 118. [Google Scholar] [CrossRef] - Lin, S.; Shao, L.; Hui, C.; Song, Y.; Reddy, G.V.P.; Gielis, J.; Li, F.; Ding, Y.; Wei, Q.; Shi, P. Why does not the leaf weight-area allometry of bamboos follow the 3/2-power law? Front. Plant Sci.
**2018**, 9, 583. [Google Scholar] [CrossRef] [PubMed] - Shi, P.; Huang, J.; Hui, C.; Grissino-Mayer, H.D.; Tardif, J.; Zhai, L.; Li, B. Capturing spiral radial growth of conifers using the super ellipse to model tree-ring geometric shape. Front. Plant Sci.
**2015**, 6, 856. [Google Scholar] [CrossRef] [PubMed] - Thompson, D.W. On Growth and Form; Cambridge University Press: London, UK, 1917. [Google Scholar]
- O’Shea, B.; Mordue-Luntz, A.J.; Fryer, R.J.; Pert, C.C.; Bricknell, I.R. Determination of the surface area of a fish. J. Fish. Dis.
**2006**, 29, 437–440. [Google Scholar] [CrossRef] [PubMed] - Shi, P.; Ishikawa, T.; Sandhu, H.S.; Hui, C.; Chakraborty, A.; Jin, X.; Tachihara, K.; Li, B. On the 3/4-exponent von Bertalanffy equation for ontogenetic growth. Ecol. Model.
**2014**, 276, 23–28. [Google Scholar] [CrossRef] - Firman, D.M.; Allen, E.J. Estimating individual leaf area of potato from leaf length. J. Agric. Sci.
**1989**, 112, 425–426. [Google Scholar] [CrossRef] - Hastie, T.; Tibshirani, R.; Friedman, J. The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed.; Springer: Berlin, Germany, 2009. [Google Scholar]
- Shi, P.; Chen, Z.; Reddy, G.V.P.; Hui, C.; Huang, J.; Xiao, M. Timing of cherry tree blooming: Contrasting effects of rising winter low temperatures and early spring temperatures. Agric. For. Meteorol.
**2017**, 240–241, 78–89. [Google Scholar] [CrossRef] - Efron, B.; Tibshirani, R.J. An Introduction to the Bootstrap; Chapman and Hall/CRC: New York, NY, USA, 1993. [Google Scholar]
- Sandhu, H.S.; Shi, P.; Kuang, X.; Xue, F.; Ge, F. Applications of the bootstrap to insect physiology. Fla. Entomol.
**2011**, 94, 1036–1041. [Google Scholar] [CrossRef]

**Figure 1.**A leaf of one bamboo plant and fitted results using the simplified Gielis equation (SGE). (

**a**) Scanned leaf image; (

**b**) Observed and predicted leaf edge. The gray curve represents the scanned (observed) leaf edge, and the red curve represents the leaf edge predicted by the SGE.

**Figure 2.**Leaf examples of four tree species. (

**a**) Pittosporum tobira (Thunberg) W. T. Aiton; (

**b**) Chimonanthus praecox (Linnaeus) Link; (

**c**) Aucuba japonica var. variegata Dombrain; (

**d**) Phoebe sheareri (Hemsl.) Gamble.

**Figure 3.**Leaf examples of 10 geographical populations of Parrotia subaequalis in eastern China. The letters below the leaf photos represent the codes of the geographical populations. AJ: Anji, Zhejiang Province; HN: Xinyang, Henan Province; HW: Huangwei, Yuexi, Anhui Province; HZ: Changhua, Huangzhou, Zhejiang Province; JD: Jingde, Anhui Province; JS: Yixing, Jiangsu Province; SC: Shucheng, Anhui Province; TC: Tongcheng, Anhui Province; TT: Tiantangzhai, Jinzhai, Anhui Province; TX: Tianxia, Yuexi, Anhui Province.

**Figure 4.**Fitted results using the SGE for four tree species. Panels (

**a**–

**d**) represent P. tobira, C. praecox, A. japonica var. variegata and P. sheareri, respectively. For every panel, the gray curve represents the scanned (observed) leaf edge, and the red curve represents the leaf edge predicted by the SGE. The leaves correspond to those in Figure 2.

**Figure 5.**Fitted results using the SGE for the 10 geographical populations of P. subaequalis. AJ: Anji, Zhejiang Province; HN: Xinyang, Henan Province; HW: Huangwei, Yuexi, Anhui Province; HZ: Changhua, Huangzhou, Zhejiang Province; JD: Jingde, Anhui Province; JS: Yixing, Jiangsu Province; SC: Shucheng, Anhui Province; TC: Tongcheng, Anhui Province; TT: Tiantangzhai, Jinzhai, Anhui Province; TX: Tianxia, Yuexi, Anhui Province. For every panel, the gray curve represents the scanned (observed) leaf edge, and the red curve represents the leaf edge predicted by the SGE. The leaves correspond to those in Figure 3.

**Figure 6.**Comparison between the scanned leaf area and the predicted leaf area based on leaf length and leaf width with an additional parameter, i.e., the floating ratio in leaf length (c). Panels (

**a**–

**i**) represent the results of the comparison between the scanned and predicted leaf areas for the nine datasets (see 2.1 Materials for details), respectively. MPE denotes the mean percent error of leaf area. The solid line represents y = x, and the dashed line represents the regression line between the scanned and the predicted leaf areas. In each panel, the small open circles regardless of colors represent the planar coordinates consisting of the scanned leaf area (as the x-coordinate) and the predicted leaf area (as the y-coordinate). In panels (

**a**,

**b**,

**h**), each color represents a species. However, the same colors in different panels do not represent the same species. In panel (

**i**), different colors represent different geographical populations.

**Figure 7.**Effect of the mean percent error of leaf length on the mean percent error of leaf area based on the data of (

**a**) Phyllostachys incarnata and (

**b**) Pleioblastus chino (i.e., datasets 4 and 5). The grey solid circles represent the planar coordinates consisting of the mean percent error of leaf length (MPEL, as x-coordinate) and the mean percent error of leaf area (MPEA, as y-coordinate); the red straight line came from the equation y = 1/2 x. Take a grey solid circle whose MPEL equals 20% in panel (

**a**), for example. There are 209 leaves for Ph. incarnata whose observations of leaf length and area are both known. For each leaf, we redefine its length by randomly choosing a value in the range (0.8 L, 1.2 L), where L denotes the actual observation of leaf length. Then we can calculate the predicted value of leaf area based on the simplified Gielis equation, and consequently we are able to calculate the percent error of leaf area for this leaf. Because there are 290 leaves, we can obtain the mean percent error of leaf area that is exactly the y-coordinate associated with MPEL = 20% (namely the x-coordinate).

**Figure 8.**Scaling relationship between leaf area and leaf length. (

**a**) Bambusa multiplex f. fernleaf (R. A. Young) T. P. Yi; (

**b**) Indocalamus barbatus McClure; (

**c**) Indocalamus pedalis (Keng) P. C. Keng; (

**d**) Indocalamus pumilus Q. H. Dai and C. F. Keng; (

**e**) Indocalamus victorialis P. C. Keng; (

**f**) Phyllostachys incarnata T. H. Wen; (

**g**) Pleioblastus chino (Franchet & Savatier) Makino; (

**h**) Pseudosasa amabilis (McClure) Keng f.; (

**i**) Sasaella kongosanensis ‘Aureostriatus’ (Nakai) Nakai ex Koidz. The open circles represent observations, and the solid line represents the regression line between leaf area and leaf length. Here, $\widehat{b}$ represents the estimate of slope (i.e., the scaling exponent between leaf area and leaf length); R

^{2}represents the coefficient of determination to reflect the goodness of fit; CV is the coefficient of variation of the ratios of leaf width to leaf length; S

_{k}represents the skewness of the ratios; μ represents the mean of the ratios.

**Figure 9.**Comparison of the goodness of fit for two methods. Three datasets, each representing a species, were used. The GAM represents the generalized additive model, and the SGE represents the simplified Gielis equation. CV represents the coefficient of variation in root mean squared error (RMSE). To test whether there is a significant difference in RMSE between the two methods, 3000 partitionings of the dataset were carried out using the golden ratio (38.2% for the training set and 61.8% for the test set) for each species’ dataset. Boxplots were obtained, based on these 3000 partitionings. Panels (

**a**–

**c**) represent the training sets, and panels (

**d**–

**f**) represent the test sets. Panels (

**g**–

**i**) exhibit the distribution of the differences in RMSE between the two methods based on the test set. The 95% confidence interval (CI) for each panel includes 0, which means that there was no significant difference in the goodness of fit between the two methods in predicting the leaf areas of the test set (see ref. [26]). However, there was an apparent significant difference in RMSE between the two methods based on the training set. The vertical blue dashed line represents the mean of the differences, and the two vertical gray dashed lines represent the 95% CI of the difference.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shi, P.; Ratkowsky, D.A.; Li, Y.; Zhang, L.; Lin, S.; Gielis, J.
A General Leaf Area Geometric Formula Exists for Plants—Evidence from the Simplified Gielis Equation. *Forests* **2018**, *9*, 714.
https://doi.org/10.3390/f9110714

**AMA Style**

Shi P, Ratkowsky DA, Li Y, Zhang L, Lin S, Gielis J.
A General Leaf Area Geometric Formula Exists for Plants—Evidence from the Simplified Gielis Equation. *Forests*. 2018; 9(11):714.
https://doi.org/10.3390/f9110714

**Chicago/Turabian Style**

Shi, Peijian, David A. Ratkowsky, Yang Li, Lifang Zhang, Shuyan Lin, and Johan Gielis.
2018. "A General Leaf Area Geometric Formula Exists for Plants—Evidence from the Simplified Gielis Equation" *Forests* 9, no. 11: 714.
https://doi.org/10.3390/f9110714