^{1}

^{1}

^{2}

^{3}

^{4}

^{5}

^{1}

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^{1}

^{2}

^{3}

^{4}

^{5}

Edited by: Jian-Guo Huang, South China Botanical Garden (CAS), China

Reviewed by: Ze-Xin Fan, Xishuangbanna Tropical Botanical Garden (CAS), China; Chiara Cirillo, University of Naples Federico II, Italy; Roland Valcke, University of Hasselt, Belgium

This article was submitted to Functional Plant Ecology, a section of the journal Frontiers in Plant Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The principle of similarity (Thompson, ^{2} and that ^{2}. This then allowed us to derive the density-thickness allometry ρ ∝ ^{b} and the weight-area allometry ^{(b+3)/2} ≈ ^{9/8}, where

The leaf of a plant is an important organ for transpiration, photosynthesis and heat balance (Parkhurst and Loucks,

Leaf dry weight (^{β}, where α is the normalization constant and β the scaling exponent (Niklas et al.,

Bamboos, the subfamily Bambusoideae of the grass family Poaceae (or Graminaceae), have been widely used for food, handicraft and construction materials. It includes 75 genera and 1,300 species, covering 25 million hectares worldwide (Liese and Köhl,

We measured leaf surface area and leaf biomass values of at least 100 leaves for each of the 11 bamboo species located in the Nanjing Forestry University campus used in this study (Table

Bamboo species, sampling time and sample size of leaves.

1 | Early June, 2016 | 200 | |

2 | Early May, 2016 | 200 | |

3 | Late April, 2016 | 200 | |

4 | Late April, 2016 | 200 | |

5 | Mid-September, 2016 | 200 | |

6 | Mid-September, 2016 | 200 | |

7 | Early December, 2016 | 210 | |

8 | Early July, 2014 | 112 | |

9 | Early July, 2014 | 108 | |

10 | Early July, 2014 | 113 | |

11 | Early July, 2014 | 121 |

To connect these leaf functional traits, let

Here

Assuming that parameter

(see Appendix for mathematical proofs).

Comparison between the scanned and predicted leaf edges of

By definition, we have

According to the principle of similarity (Thompson,

The above equation will be tested using the experimental data (see below for details). Let us assume that leaf density and thickness follow the following relationship:

Alternatively, this can be described as:

Here, α, β,

where

If leaf density was not related to leaf thickness, ^{3/2}, directly following the principle of similarity (Thompson,

In the aforementioned derivation, the key assumption is that leaf area is proportional to the 2-power of leaf thickness (i.e., Equation 5). In order to examine this hypothesis, we measured the leaf thickness of 100 leaves of

An illustration of how the mean thickness of a bamboo leaf is measured. The yellow points represent the locations for measuring the thickness. The data on the main vein were neglected because the values are extremely higher than those apart from the main vein.

The scaling relationship was demonstrated using the reduced major axis (RMA; Milla and Reich,

The raw data used in the present study of leaf area, weight and thickness can be found in the Supplementary Material.

Using 1864 data points, we obtained ln(ρ) = −5.00 −0.736 ln(^{2} = 0.94 (Figure

The linear fitting between the natural logarithm of the proportionality to leaf thickness and that of leaf density. Different colors represent different bamboo species, and there are totally 11 species. The proportionality of leaf thickness was obtained by ^{1/2}, and the proportionality of leaf density was obtained by ^{−3/2}.

Using the leaf area and thickness data of 100 leaves of ^{2} = 0.85 (Figure

The linear fitting between the natural logarithm of leaf mean thickness and that of leaf area.

Using 1864 data points, we obtained ln(^{2} = 0.99 (Figure

The linear fitting between the natural logarithm of leaf area and that of leaf fresh weight. Different colors represent different bamboo species, and there are totally 11 species.

In this work, a negative relationship between leaf thickness and density was found in 11 bamboo species, consistent with the results for sclerophytes, mesophytes and succulents where the slope ranges from −0.46 to −0.82 (Vendramini et al.,

The possibility of the negative correlation between leaf thickness and density could stabilize LMA. Identifying the contributions of leaf thickness (

The correlation between leaf thickness and leaf area has been poorly studied. Niklas et al. (

Because the scaling exponent governing the relationship between leaf dry and fresh weight across species is statistically indistinguishable from unity (Figure _{F}) rather than dry weight (_{D}):

Comparison between the observed (points) and predicted fresh weights by using dry weights (the straight line) of _{F}) and dry weight (_{D}). The estimate of slope is 2.0830 ± 0.0178, and ^{2} = 0.9979. That is, there is a strong proportional relationship between fresh and dry weights.

Price and Enquist (^{2} = 0.99 using 1864 data points. The 95% confidence interval of slope is (1.143, 1.152) based on 3000 bootstrap replications. Our scaling exponent 1.147 is in accordance with those reported in other studies (Niklas et al.,

For bamboos, leaf area was demonstrated to be proportional to leaf mean thickness squared and also to be proportional to leaf length squared. Consequently, leaf volume can be expressed by leaf area to the power 3/2, which means that leaf volume, leaf area, leaf mean thickness (or leaf length) follow the similarity of principle of Thompson. The allometric relationship between leaf weight and leaf area based on the experimental data of 11 bamboo species was demonstrated to significantly deviate from the 3/2–power law. In other words, leaf weight is not proportional to leaf volume because leaf volume is the proportionality of leaf area to the power 3/2. To find the reason of the deviation to the 3/2–power law, we analyzed the relationship between the proportionality of leaf density and that of leaf mean thickness, and found a significant negative linear relationship between the log-transformed data of these two variables. That is, leaf density decreases with leaf size (represented by leaf mean thickness or leaf length or leaf area) increasing, which answers why leaf weight does not have a proportional relationship with leaf area to the power 3/2. The main conclusion in the present study that the negative scaling exponent of leaf thickness to density results in the deviation of the scaling exponent of leaf area to weight from the 3/2–power law could be potentially extended to other plants. However, the negative scaling exponent of leaf thickness to density (as demonstrated to be approximate to −3/4 for bamboos) might vary with different species of plants. In addition, whether the leaf area-thickness allometry follows the 2–power law for other plants also merits further investigation.

SL and PS designed the experiment; LS, FL, YS, QW, and YD carried out this experiment; PS analyzed the data; PS, SL, CH, JG, and GR wrote the manuscript. All authors read and commented on this manuscript.

The reviewer Z-XF declared a shared affiliation, with no collaboration, with one of the authors YS, to the handling Editor. The other authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are thankful to two reviewers and the handling editor for their constructive and valuable comments on the earlier version of this paper. This work was financially supported by the National Key Research & Development Program of China (2016YFD0600901); Jiangsu Province Support Project (LYSX [2016]04; BE2016304); the National Natural Science Foundation for Young Scholars of China (31000294) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

The Supplementary Material for this article can be found online at:

where “Gamma” represents the gamma function, and “HypergeometricPFQ” represents the generalized hypergeometric function. The detailed definitions for these two functions can be found in Wolfram MathWorld (

Leaf length has demonstrated to be proportional to parameter

Then we have: