Proportional Relationship Between Leaf Area and the Product of Leaf Length and Width of Four Types of Special Leaf Shapes

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Article Proportional Relationship between Leaf Area and the Product of Leaf Length and Width of Four Types of Special Leaf Shapes

Peijian Shi 1 , Mengdi Liu 1, Xiaojing Yu 1,*, Johan Gielis 2 and David A. Ratkowsky 3 1 Co-Innovation Centre for Sustainable Forestry in Southern China, College of Biology and the Environment, Bamboo Research Institute, Nanjing Forestry University, Nanjing 210037, China; [email protected] (P.S.); [email protected] (M.L.) 2 Department of Biosciences Engineering, University of Antwerp, B-2020 Antwerp, Belgium; [email protected] 3 Tasmanian Institute of Agriculture, University of Tasmania, Private Bag 98, Hobart, Tasmania 7001, Australia; [email protected] * Correspondence: [email protected]

Received: 22 December 2018; Accepted: 18 February 2019; Published: 19 February 2019

Abstract: The leaf area, as an important leaf functional trait, is thought to be related to leaf length and width. Our recent study showed that the Montgomery equation, which assumes that leaf area is proportional to the product of leaf length and width, applied to different leaf shapes, and the coefficient of proportionality (namely the Montgomery parameter) range from 1/2 to π/4. However, no relevant geometrical evidence has previously been provided to support the above findings. Here, four types of representative leaf shapes (the elliptical, sectorial, linear, and triangular shapes) were studied. We derived the range of the estimate of the Montgomery parameter for every type. For the elliptical and triangular leaf shapes, the estimates are π/4 and 1/2, respectively; for the linear leaf shape, especially for the plants of Poaceae that can be described by the simplified Gielis equation, the estimate ranges from 0.6795 to π/4; for the sectorial leaf shape, the estimate ranges from 1/2 to π/4. The estimates based on the observations of actual leaves support the above theoretical results. The results obtained here show that the coefficient of proportionality of leaf area versus the product of leaf length and width only varies in a small range, maintaining the allometric relationship for leaf area and thereby suggesting that the proportional relationship between leaf area and the product of leaf length and width broadly remains stable during leaf evolution.

Keywords: allometric relationship; leaf area formula; linear regression; the Montgomery equation; the simpliﬁed Gielis equation

1. Introduction The leaf is the most important photosynthetic organ of plants. Leaves participate in many physiological processes that sometimes result in trade-offs among carbon fixation, water loss, and defense from herbivore attack. There are clearly multiple solutions or ways to optimize leaf form, as evidenced by the wide variety of leaves that can be found in almost any environment [1]. During long-term evolution, plants evolved different leaf shapes to enable them to survive and win the fierce interspecific competition in plant communities [2]. The leaves of plants exhibit a large variation in shape, ranging from an elliptic leaf to a palmate leaf. To calculate the leaf area of corn, Montgomery [3] proposed a formula that assumes that leaf area is proportional to the product of leaf length and width. We call this formula the Montgomery equation in the present study and we refer to the coefficient of proportionality in the equation as the Montgomery parameter. This equation was

Forests 2019, 10, 178; doi:10.3390/f10020178 www.mdpi.com/journal/forests Forests 2019, 10, 178 2 of 13 confirmed to hold for crops with different leaf shapes, including the castor bean (Ricinus communis L.) [4,5]. However, the Montgomery equation had been largely neglected in the investigation of the leaves of woody plants. Our recent study showed that the Montgomery equation was also able to describe the proportional relationship between leaf area and the product of leaf length and width of six classes of plants: 10 geographical populations of Parrotia subaequalis (H. T. Chang) R. M. Hao & H. T. Wei, two species of tulip trees with their hybrid, 12 species of Bambusoideae, five species of Lauraceae, five species of Oleaceae, and 12 species of Rosaceae, where 150−500 leaves were used for each population or each species. The estimates of the Montgomery parameter all fell into the range (1/2, π/4) [6]. If the Montgomery equation did apply to all broad-leaved plants and the estimates of the Montgomery parameter for different plants fell into a narrow range, it would imply that the allometric relationship between leaf area and the product of leaf length and width maintains a certain stability during leaf evolution. However, the relevant geometric derivation associated with leaf shape has not been studied. In this study, we investigated four leaf shapes whose areas can be described by mathematical equations to derive theoretical or limiting values of the Montgomery parameter. In addition, to further test the validity of the theoretical derivation, the leaves of four species of plants with similar leaf shapes to the above mathematical equations were used.

2. Materials and Methods

2.1. Leaf Area Models Associated with the Montgomery Parameter Let A, L, and W represent leaf area, leaf length, and leaf width, respectively. The Montgomery equation is A = kLW, where k is a parameter to be ﬁtted, referred to here as the Montgomery parameter, and which can be rewritten as: A k = (1) LW (1) For an elliptical leaf, its semi-major axis is half its leaf length, and its semi-minor axis is half its leaf width. Thus, the area of an ellipse can be expressed as:

L W π A = π = LW (2) 2 2 4 In this case, the Montgomery parameter equals π/4. (2) For a sectorial leaf, we use the radius from leaf base to the farthest edge to represent its length, the straight-line distance of two apexes of the arc to represent its width, and θ to represent the arc angle in radians. Then, leaf width W = 2 L sin(θ/2). Because A = L2 θ/2 (i.e., the formula for the area of a sector of a circle), we have, according to Equation (1):

A 1 θ/2 k = = (3) LW 2 sin(θ/2)

For a sectorial leaf, θ lies in the range (0, π). Note that in the case of θ > π (e.g., Nymphaea tetragona Georgi), it is unsuitable to define leaf length as the radius of a sector, and leaf length should be defined 1 θ/2 as twice the radius. When θ = π, k = π/4. It is easy to prove limk = because lim ( ) = 1. Thus, θ→0 2 θ/2→0 sin θ/2 for a sectorial leaf (on the condition that the arc angle does not exceed π), the Montgomery parameter ranges from 1/2 to π/4. (3) For a bamboo leaf, it can be described by the simplified Gielis equation (SGE) [7–10]:

l r(ϕ) = (4) ϕ ϕ 1/n cos 4 + sin 4 Forests 2019, 10, 178 3 of 13 where r and ϕ represent the polar radius and polar angle, respectively; l and n are the parameters to be ﬁtted. For a leaf, ϕ ranges from 0 to 2π. Assume that the polar radius and polar angle associated with leaf width in the ﬁrst quadrant are rw and α. Leaf length and width have been demonstrated by Shi et al. [8] to be: L = 1 + 2−0.5/n l (5)

and W = 2(sin α)rw = 2 sin α L (6) −0.5/n α α 1/n (1+2 )(cos 4 +sin 4 ) There is a special relationship between n and α, which is [8]:

α α tan α cos 4 − sin 4 n = α α (7) 4 cos 4 + sin 4

Leaf area is equal to [8,10]:

1 R 2π 2 1 R 2π 2 ϕ ϕ −2/n A = 2 0 r dϕ = 2 0 l cos 4 + sin 4 dϕ 2 −2/n (8) = 1 L R 2π cos ϕ + sin ϕ dϕ 2 1+2−0.5/n 0 4 4

R 2π −2/n Here, 0 {cos(ϕ/4) + sin(ϕ/4)} dϕ was demonstrated to be a function of n [10]. Assuming that leaf length is constant, we have:

2R 2π ϕ ϕ −2/n 1 L (cos +sin ) dϕ = A = 2 1+2−0.5/n 0 4 4 k LW L2 2 sin α 1/n 1+2−0.5/n cos α +sin α (9) ( )( 4 4 ) 1/n (cos α +sin α ) −2/n = 1 1 4 4 R 2π cos ϕ + sin ϕ dϕ 4 sin α 1+2−0.5/n 0 4 4

From the above, we derive limk ≈ 0.6795 and lim k = π/4. In other words, for leaves with the α→0 α→π/2 SGE leaf shape, the Montgomery parameter ranges from 0.6795 to π/4. (4) For a triangular leaf, let its height represent its length and its base represent its width. According to the triangular area formula, its area is equal to

1 A = LW (10) 2 In this case, the Montgomery parameter equals 1/2.

2.2. Materials More than 390 leaves of Hydrocotyle vulgaris L. were collected at the Xuanwu Lakeside Park, Nanjing, China, as the representative of an elliptical shape; more than 380 leaves of Ginkgo biloba L. were collected in the Nanjing Forestry University campus, Nanjing, China, as representative of a sector shape; 250−500 leaves of each of 20 bamboo species were collected in the Nanjing Forestry University campus (see Table S1 in the online Supplementary Materials for details), and among these bamboo species more than 300 leaves of Oligostachyum sulcatum Z.P. Wang et G.H. Ye were used as an example representative of a linear shape; more than 340 leaves of Polygonum perfoliatum L. were collected in the White Horse experimental station of Nanjing Forestry University as the representation of the triangular shape. The detailed collection information of the four representative species is listed in Table1. Figure1 shows the intuitive examples for the four types of leaf shapes. 2.2. Materials More than 390 leaves of Hydrocotyle vulgaris L. were collected at the Xuanwu Lakeside Park, Nanjing, China, as the representative of an elliptical shape; more than 380 leaves of Ginkgo biloba L. were collected in the Nanjing Forestry University campus, Nanjing, China, as representative of a sector shape; 250−500 leaves of each of 20 bamboo species were collected in the Nanjing Forestry University campus (see Table S1 in the online Supplementary Material for details), and among these bamboo species more than 300 leaves of Oligostachyum sulcatum Z.P. Wang et G.H. Ye were used as an example representative of a linear shape; more than 340 leaves of Polygonum perfoliatum L. were collected in the White Horse experimental station of Nanjing Forestry University as the representation of the triangular shape. The detailed collection information of the four representative species is listed in Table 1. Figure 1 shows the

Forestsintuitive2019, 10 ,examples 178 for the four types of leaf shapes. 4 of 13

Figure 1. Examples of the four types of leaf shapes.

Table 1. Collection informationFigure of 1. fourExamples representative of the four leaf types shapes. of leaf All shapes. leaves were collected in Nanjing city, Jiangsu Province, China, in 2018.

Family Latin Name Collection Location Sampling Time Sample Size Araliaceae Hydrocotyle vulgaris L. 32◦40900 N, 118◦4802300 E 1 December 2018 393 Ginkgoaceae Ginkgo biloba L. 32◦405700 N, 118◦4803400 E 28 November 2018 388 Poaceae Oligostachyum sulcatum Z.P. Wang et G.H. Ye 32◦404700 N, 118◦490200 E 21 June 2018 303 Polygonaceae Polygonum perfoliatum L. 31◦3601900 N, 119◦1003500 E 30 September 2018 348

2.3. Leaf Image Processing and Data Acquisition Each leaf was scanned to a bitmap image in a resolution of 400−600 dpi, and the magic wand in Photoshop (version ≥CS3) was used to extract the leaf edge. The edge was saved in a new layer. We filled the outer part of the leaf edge with black and filled the inner part of the leaf edge with white, which is beneficial for the extraction of the planar coordinates of the leaf edge [11]. This new layer was referred to as the black-white image. Then we used an M file in Matlab (version ≥2009a) developed by Shi et al. [8] to extract the planar coordinates of the leaf edge and the number of data points relied on the resolution of the black-white image. Finally, we used the R (version ≥3.2.2) [12] script developed by Shi et al. [8] to obtain the measures of leaf morphologic parameters, such as leaf area, length, width, and perimeter. If the leaf surface is sufficiently flat, measured leaf parameters will be very accurate. However, this method does not apply to plants like Ilex cornuta Lindl. et Paxt. that do not have flat leaves. Leaf length was defined as the distance from leaf apex to the junction of the lamina and petiole of a leaf. We defined the straight line through these two points as the x-axis. Then we set up 1000 strips of equal width perpendicular to the x-axis starting with the leaf apex and ending at the junction of Forests 2019, 10, 178 5 of 13 the lamina and petiole. The leaf width was defined to be the widest part of the leaf perpendicular to the x-axis.

2.4. Statistical Analysis For each species, assume that A represents leaf area, L represents leaf length, and W represents leaf width. To stabilize the variance, we used the log-transformation of leaf area and that of the product between leaf length and width. Then we carried out the following linear regression:

ln(A) = a + ln(LW) (11) where a is a parameter to be fitted. In this case, the Montgomery parameter is equal to exp(a). Then the linear regression with slope equal to one was carried out to estimate the intercept and its 95% confidence interval (CI). We also fitted the pooled data of 20 bamboo species to check whether the pooled data still could satisfy the Montgomery equation. Considering that the leaf shapes of these plants exhibited a large difference, we also used Tukey’s HSD (Honestly Significant Difference) test to carry out the multiple comparisons in the ratios of leaf width to length and the ratios of leaf perimeter to area among four species of plants [13].

3. Results Four groups of data all confirm the validity of the Montgomery equation. Leaf area is demonstrated to be proportional to the product of leaf length and width; the goodness of fit for each data set is high (r > 0.985; Figure2). The estimate of the Montgomery parameter for H. vulgaris is 0.7673, which is slightly lower than π/4 (Figure2a). This means that the leaf shape of H. vulgaris slightly deviates from a standard ellipse. The estimate of the Montgomery parameter of G. biloba is equal to 0.6228, which lies in the range (1/2, π/4) derived from a standard sector (Figure2b). The estimate of the Montgomery parameter of O. sulcatum is equal to 0.7196, which also falls into the range (0.6795, π/4) derived from a standard SGE shape (Figure2c). Actually, the estimates of the Montgomery parameter for the 20 bamboo species all fall into that range (Figure3). Even for the pooled data of 20 bamboo species, the Montgomery equation still validly describes the proportionality of the product of leaf length and width to leaf area, and the estimate of the Montgomery parameter equals 0.7141 (Figure4). The estimate of the Montgomery parameter of P. perfoliatum equals 0.5497, which is slightly higher than 0.5 derived from a standard triangle (Figure2d). The 95% confidence intervals corresponding to the aforementioned estimates of the Montgomery parameter are also all between 1/2 and π/4. Overall, all of the estimates lie within the range (1/2, π/4). There is a significant difference in the ratio of leaf width to length among the four types of plants. The ratio of leaf width to length of P. perfoliatum is significantly higher than the others, and that of the bamboo is the smallest (Figure5a). There is also a significant difference in the ratio of leaf perimeter to area among the four types of plants. The ratio of leaf perimeter to area of O. sulcatum is significantly higher than the others, and that of the ginkgo is the smallest (Figure5b). Forests 2018, 9, x FOR PEER REVIEW 2 of 14

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46 47 Figure 2. Linear regression of the Montgomery equation for four species of plants. (a) H. vulgaris; (b) 4846 G. biloba; (c) O. sulcatum; (d) P. perfoliatum. In each panel, A denotes leaf area; L represents leaf length; 4947 FigureWFigure represents 2.2. LinearLinear leaf regression width; RMSE of the the repr Montgomery Montgomeryesents the root-mean-squareeq equationuation for for four four specieserror; species r ofrepresents of plants. plants. (a )the( aH.) H.correlationvulgaris vulgaris; (b; ) 5048 (coefficient;bG.) bilobaG. biloba; (c )n;( O. representsc )sulcatumO. sulcatum; (thed) P.;( number perfoliatumd) P. perfoliatum of . leavesIn each. sampled; Inpanel, each A panel, denotesexp( aA) leafdenotesrepresents area; leafL representsthe area; estimateL representsleaf oflength; the 5149 leafMontgomeryW represents length; W parameter;representsleaf width; leafRMSE95% width; CI repr denotes RMSEesents representsthethe root-mean-square95% confidence the root-mean-square intervalerror; r representsof error;the estimater representsthe correlation of the 5250 correlationMontgomerycoefficient; coefficient;n parameter. representsn Openrepresents the circlesnumber the represent of number leaves the ofsampled; observations, leaves sampled; exp( anda ) represents exp(the redaˆ) represents straight the estimateline the represents estimate of the 5351 oftheMontgomery the fitted Montgomery regression parameter; line. parameter; 95% 95%CI denotes CI denotes the the95% 95% confidence confidence interval interval of of the the estimate ofof thethe 52 MontgomeryMontgomery parameter. parameter. Open Open circles circles represent represent the the observations, observations, and and the the red red straight straight line line represents represents 53 thethe fitted fitted regression regression line. line.

54 55 Figure 3. The estimates of the Montgomery parameters (MP) and the correlation coefficients coefﬁcients for 20 5654 bamboo species. Blue open circles represent the esti estimatesmates of the MP; and red open triangles represent 5755 theFigure correlation 3. The coefﬁcients.coefficients.estimates of The Thethe numberMontgomerynumber on on the the parametersx -axisx-axis represents represents (MP) theand the bamboo the bamboo correlation species species codecoefficients code (see (see Table Tablefor S120 5856 inS1bamboo thein the online onlinespecies. Supplementary Supplementa Blue open circles Materialsry Materials represent for for details). thedetails). estimates of the MP; and red open triangles represent 57 the correlation coefficients. The number on the x-axis represents the bamboo species code (see Table 58 S1 in the online Supplementary Materials for details).

Forests 2018, 9, x FOR PEER REVIEW 3 of 14

Forests 2019, 10, 178 7 of 13 Forests 2018, 9, x FOR PEER REVIEW 3 of 14

59 60 Figure 4. Linear regression of the Montgomery equation for the pooled data of 20 bamboo species. A 61 denotes leaf area; L represents leaf length; W represents leaf width; RMSE represents the root-mean- 62 square error; r represents the correlation coefficient; n represents the number of leaves sampled; exp(a) 63 represents the estimate of the Montgomery parameter; 95% CI denotes the 95% confidence interval of 64 the estimate of the Montgomery parameter. Open circles (among which different colors represent 65 different bamboo species) represent the observations, and the solid straight line represents the fitted 6659 regression line. Figure 4. Linear regression of the Montgomery equation for the pooled data of 20 bamboo species. 60 Figure 4. Linear regression of the Montgomery equation for the pooled data of 20 bamboo species. A A denotes leaf area; L represents leaf length; W represents leaf width; RMSE represents the 6761 Theredenotes is leafa significant area; L represents difference leaf in length; the ratio W represents of leaf width leaf width;to length RM amongSE represents the four the types root-mean- of plants. root-mean-square error; r represents the correlation coefficient; n represents the number of leaves 6862 The ratiosquare of error; leaf widthr represents to length the correlation of P. perfoliatum coefficient; is nsignificantly represents the higher number than of leaves the others,sampled; and exp( thata) of sampled; exp(aˆ) represents the estimate of the Montgomery parameter; 95% CI denotes the 95% 6963 the bamboorepresents is thethe estimate smallest of the(Figure Montgomery 5a). There paramete is alsor; 95% a significantCI denotes thedifference 95% confidence in the interval ratio ofof leaf confidence interval of the estimate of the Montgomery parameter. Open circles (among which different 7064 perimeterthe estimate to area of among the Montgomery the four types parameter. of plan Opents. The circles ratio (among of leaf perimeterwhich different to area colors of O. represent sulcatum is colors represent different bamboo species) represent the observations, and the solid straight line 7165 significantlydifferent higher bamboo than species) the others,represent and the that observations, of the ginkgo and the is thesolid smallest straight (Figureline represents 5b). the fitted represents the fitted regression line. 66 regression line.

67 There is a significant difference in the ratio of leaf width to length among the four types of plants. 68 The ratio of leaf width to length of P. perfoliatum is significantly higher than the others, and that of 69 the bamboo is the smallest (Figure 5a). There is also a significant difference in the ratio of leaf 70 perimeter to area among the four types of plants. The ratio of leaf perimeter to area of O. sulcatum is 71 significantly higher than the others, and that of the ginkgo is the smallest (Figure 5b).

72 73 FigureFigure 5.5. ComparisonComparison of the ratios of leafleaf widthwidth toto lengthlength (and(and thethe ratiosratios ofof leafleaf perimeterperimeter to area)area) 74 amongamong thethe fourfour speciesspecies ofof plants.plants. ((aa)) TheThe ratioratio ofof leafleaf widthwidth toto length;length; ((bb)) thethe ratioratio ofof leafleaf perimeterperimeter toto 75 area.area. InIn panelpanel ((aa),), thethe redred percentagepercentage numbernumber aboveabove thethe boxbox isis thethe coefficientcoefficient ofof variationvariation ofof thethe ratiosratios 76 ofof leafleaf widthwidth toto length;length; thethe greengreen numbernumber aboveabove thethe boxbox isis thethe skewnessskewness ofof thethe ratiosratios ofof leafleaf widthwidth toto 77 length.length. TheThe lettersletters A,A, B,B, C,C, andand DD in each panel were used to indicate a significantsignificant difference ofof speciesspecies 78 meansmeans usingusing Tukey’sTukey’s HSDHSD ((αα= = 0.05).0.05). DataData setset 11 representsrepresents H.H. vulgarisvulgaris;; datadata setset 22 representsrepresents G.G. bilobabiloba;; 7972 datadata setset 33 representsrepresentsO. O. sulcatumsulcatum;; datadata set set 4 4 represents representsP. P. perfoliatum perfoliatum. . 73 4. DiscussionFigure 5. Comparison of the ratios of leaf width to length (and the ratios of leaf perimeter to area) 8074 4. Discussionamong the four species of plants. (a) The ratio of leaf width to length; (b) the ratio of leaf perimeter to 75 4.1. Thearea. Allometric In panel Relationship(a), the red percentage between Leaf number Area above and Length the box is the coefficient of variation of the ratios 8176 4.1. Theof leafAllometric width to Relationship length; the greenbetween number Leaf Area above and the Length box is the skewness of the ratios of leaf width to Thompson stated that the area of an object is proportional to its length squared [14]. However, 77 length. The letters A, B, C, and D in each panel were used to indicate a significant difference of species 82 for plantThompson leaves, stated this principle that the doesarea notof an always object holdis proportional [6]. Figure 6to shows its length the fittedsquared results [14]. usingHowever, the 78 means using Tukey’s HSD (α = 0.05). Data set 1 represents H. vulgaris; data set 2 represents G. biloba; 83 log-transformedfor plant leaves, datathis ofprinciple leaf area does and not length. always We ho foundld [6]. that Figure the estimates 6 shows ofthe the fitted slopes results for H. using vulgaris the 79 data set 3 represents O. sulcatum; data set 4 represents P. perfoliatum. and P. perfoliatum are approximately equal to 2, whereas the estimates of another two species especially 80 G.4. bilobaDiscussionlargely deviate from 2. According to our recent studies [6,8], we found that whether leaf area is proportional to leaf length squared mainly depends on the extent of variation of the ratio of leaf 81 width4.1. The to Allometric length and Relationship the skewness between of the Leaf ratio Area distribution and Length curve. The coefficients of variation of the ratio of leaf width to length of H. vulgaris and P. perfoliatum are lower than those of the other two plant 82 Thompson stated that the area of an object is proportional to its length squared [14]. However, 83 for plant leaves, this principle does not always hold [6]. Figure 6 shows the fitted results using the

Forests 2018, 9, x FOR PEER REVIEW 4 of 14

84 log-transformed data of leaf area and length. We found that the estimates of the slopes for H. vulgaris 85 and P. perfoliatum are approximately equal to 2, whereas the estimates of another two species 86 especially G. biloba largely deviate from 2. According to our recent studies [6,8], we found that 87 whether leaf area is proportional to leaf length squared mainly depends on the extent of variation of Forests 2019, 10, 178 8 of 13 88 the ratio of leaf width to length and the skewness of the ratio distribution curve. The coefficients of 89 variation of the ratio of leaf width to length of H. vulgaris and P. perfoliatum are lower than those of 90 speciesthe other (Figure two 5planta). Although species (Figure the extent 5a). of Although the variation the extent of the ofW/L theratio variation of H. vulgaris of the W/Lis smaller ratio of than H. 91 thatvulgaris of P. is perfoliatum smaller than, the that estimate of P. ofperfoliatum the latter, isthe closer estimate to 2 thanof the that latter of theis closer former. to We2 than believe that thatof the a 92 largerformer. absolute We believe value that of the a skewnesslarger absolute of the W/Lvalueratio of oftheH. skewness vulgaris compared of the W/L to thatratio of ofP. perfoliatumH. vulgaris 93 shouldcompared lead to to that a larger of P. deviation perfoliatum of should the estimate lead to of a the larger slope deviation from 2. The of the current estimate ﬁndings of the are slope in accord from 94 with2. The the current report findings of our previousare in accord study with [8 ].the This report means of our that previous a small study variation [8]. This and means a small that absolute a small 95 valuevariation of the and skewness a small absolute of the W/L valueratio of will the resultskewness in a of 2-power the W/L law ratio for will the leafresult area–length in a 2-power allometry. law for 96 Linthe leaf et al. area–length [10] showed allometry. that several Lin et bamboo al. [10] leavesshowed followed that several the bamboo 2-power leaves law between followed leaf the area 2-power and 97 length.law between However, leaf area for more and length. bamboo However, species, thefor leafmore area–length bamboo species, allometry the leaf appeared area–length to deviate allometry from 98 theappeared 2-power to deviate law to somefrom the degree 2-power [6]. ForlawH. to vulgarissome degreeand P.[6]. perfoliatum For H. vulgaris, their and leaf P. shapes perfoliatum appear, their to 99 beleaf more shapes bilaterally appear symmetrical,to be more bilaterally so the extents symmetr of variationical, so arethe bothextents low. of However,variation forare theboth leaves low. 100 ofHowever,G. biloba for, the the angles leaves formed of G. biloba by two, the sidesangles have formed a large by two variation sides have because a large of thevariation phenomenon because ofof 101 heterophyllythe phenomenon [15], of which heterophylly consequently [15], leadswhich to consequently a large variation leads in to the a largeW/L ratio.variation in the W/L ratio.

102 Figure 6. Linear regression of the leaf area–length allometry for the four species of plants. (a) H. vulgaris; 103 Figure 6. Linear regression of the leaf area–length allometry for the four species of plants. (a) H. (b) G. biloba;(c) O. sulcatum;(d) P. perfoliatum. In each panel, A denotes leaf area; L represents leaf length; 104 vulgaris; (b) G. biloba; (c) O. sulcatum; (d) P. perfoliatum. In each panel, A denotes leaf area; L represents RMSE represents the root-mean-square error; r represents the correlation coefﬁcient; n represents the 105 leaf length; RMSE represents the root-mean-square error; r represents the correlation coefficient; n number of leaves sampled; 95% CI denotes the 95% conﬁdence interval of the estimate of the slope. 106 represents the number of leaves sampled; 95% CI denotes the 95% confidence interval of the estimate 107 4.2. Theof the Original slope. Gielis Equation and Its Validity in Describing Leaf Shape

108 Gielis proposed a polar coordinate equation to describe the shapes of many objects [16]. Tian et al. [17] reparameterized this equation to improve the model’s close-to-linear performance during the non-linear regression [18–20]:

− 1 ϕ n2 1 ϕ n3 n1 r(ϕ) = λ cos m + sin m (12) 4 γ 4 Forests 2018, 9, x FOR PEER REVIEW 5 of 14

109 4.2. The Original Gielis Equation and Its Validity in Describing Leaf Shape 110 Gielis proposed a polar coordinate equation to describe the shapes of many objects [16]. Tian et 111 al. [17] reparameterized this equation to improve the model’s close-to-linear performance during the 112 non-linear regression [18−20]:

− 1 nn 23n1 Forests 2019, 10, 178 =+φφ1  9 of 13 r ()φλcosmm sin  (12) 44γ 

113 Here,Here,r ris is the the polar polar radius radius at at the the polar polar angle angleϕ φ;;m mis is a a positive positive integer integer that that is is used used to to control control the the 114 numbernumber of of the the angles angles of of the the shape; shape;λ λ, ,γ γ, ,n n1,1,n n22,, andandn n33 areare thethe constantsconstants toto bebe fitted.fitted. FigureFigure7 7exhibits exhibits 115 thethe fitted fitted results results using using the the Gielis Gielis equation equation for for the the leaf leaf edges edges of of three three species. species. TheThe predictedpredicted edges edges are are 116 basicallybasically the the same same as as the the actual actual leaf leaf edges, edges, and theand predicted the predicted areas areareas very are approximate very approximate to the scanned to the 117 leafscanned areas leaf (see areas Table (see S2 in Table the online S2 in the Supplementary online Supplementary Materials forMaterials details). for We details). did not We fit did the leafnot fit edge the 118 ofleafG. edge biloba of. ItG. is biloba actually. It is easy actually to describe easy to describe the arc of the a sector.arc of a We sector. only We need only to need designate to designateϕ to range φ to 119 fromrange 0 from to a certain 0 to a certain angle (notangle up (not to 2upπ) to according 2π) according to the to circular the circular (or elliptical) (or elliptical) equation, equation, which which also 120 couldalso could be simulated be simulated by the Gielisby the equation. Gielis equation. However, Ho itwever, is difficult it is fordifficult us to findfor aus suitable to find parameter a suitable 121 combinationparameter combination to make the to Gielis make equation the Gielis generate equation a sector.generate There a sector. is also There the possibilityis also the possibility that the Gielis that 122 equationthe Gielis cannot equation generate cannot a standardgenerate a sector standard or approximate sector or approximate it. Shi et al. it. [8 ]Shi stated et al. that [8] thestated simplified that the 123 Gielissimplified equation Gielis was eq ableuation to calculatewas able theto calculate leaf shapes the of leaf many shapes broad-leaved of many plantsbroad-leaved by introducing plants by a 124 floatingintroducing ratio a in floating leaf length. ratio Fromin leaf the length. current From study, the sincecurrent leaf study, areais since proportional leaf area tois proportional the productof to 125 leafthe lengthproduct and of width,leaf length keeping and leaf width, width keeping constant leaf but width increasing constant or but decreasing increasing leaf or length decreasing will make leaf 126 thelength predicted will make leaf areathe predicted more closely leaf approacharea more its closely actual approach value. its actual value.

127 Figure 7. Gielis fit to the leaf edge data of three species of plants. (a) H. vulgaris;(b) O. sulcatum; 128 Figure 7. Gielis fit to the leaf edge data of three species of plants. (a) H. vulgaris; (b) O. sulcatum; (c) P. (c) P. perfoliatum. The gray curve represents the scanned (observed) leaf edge; the red curve represents 129 perfoliatum. The gray curve represents the scanned (observed) leaf edge; the red curve represents the the predicted leaf edge by the Gielis equation; the intersection of two blue dashed lines represents the 130 predicted leaf edge by the Gielis equation; the intersection of two blue dashed lines represents the polar point of the Gielis equation; the gray dashed line represents the rotated straight line that was the 131 polar point of the Gielis equation; the gray dashed line represents the rotated straight line that was previous x-axis for a standard Gielis shape. 132 the previous x-axis for a standard Gielis shape. 4.3. The Angle of G. biloba G. biloba is a precious relic tree species that has significant economic value. It has five cultivars according to seed morphology [17,21]. Its leaf shape and leaf vein pattern have received considerable attention [22,23]. Although the leaves of G. biloba do exhibit a deviation from perfect bilateral symmetry, the areal ratios of the left side to the right side tend to be 1, which indicates that the left deviation and the right deviation from bilateral symmetry are basically the same [23]. Leigh et al. [15] compared two types of different leaves of G. biloba, short- and long-shoot leaves, and found that both the structure Forests 2018, 9, x FOR PEER REVIEW 6 of 14

133 4.3. The Angle of G. biloba 134 G. biloba is a precious relic tree species that has significant economic value. It has five cultivars 135 according to seed morphology [17,21]. Its leaf shape and leaf vein pattern have received considerable 136 attention [22,23]. Although the leaves of G. biloba do exhibit a deviation from perfect bilateral 137 symmetry, the areal ratios of the left side to the right side tend to be 1, which indicates that the left Forests 2019, 10, 178 10 of 13 138 deviation and the right deviation from bilateral symmetry are basically the same [23]. Leigh et al. [15] 139 compared two types of different leaves of G. biloba, short- and long-shoot leaves, and found that both 140 andthe structure physiology and of thesephysiology two types of these of leaves two weretypes significantly of leaves were different. significantly Niklas anddifferent. Christianson Niklas [and24] 141 studiedChristianson the leaf [24] weight–area studied the allometries leaf weight–area of 13 biologically allometries distinctof 13 biologically categories ofdistinct leaves categories of G. biloba of, 142 andleaves they of G. obtained biloba, and seven they statistically obtained seven significantly statistically different significantly models different describing models the scalingdescribing of leaf the 143 weightscaling andof leaf area. weight The current and area. study The demonstrates current study that demonstrates the Montgomery that the equation Montgomery can hold equation regardless can 144 ofhold the regardless difference of in the leaf difference shape of G.in bilobaleaf .shape However, of G. whetherbiloba. However, the Montgomery whether parametersthe Montgomery from 145 differentparameters cultivars from ordifferent geographical cultivars populations or geographical of G. biloba populationshave significant of differencesG. biloba have deserves significant further 146 investigation.differences deserves In addition, further the Montgomeryinvestigation. equation In addition, is helpful the toMontgomery quantify the equation angles of is the helpful leaves ofto 147 G.quantify biloba. the Because angles the of leaf the ofleavesG. biloba of G.is biloba not a. Because standard the sector, leaf itof createsG. biloba difficulties is not a standard for investigators sector, it 148 whocreates attempt difficulties to measure for investigators and compare who the attempt angles of to different measure cultivars and compare or geographical the angles populations. of different 149 Thecultivars angles or ofgeographical leaves might populations. also reflect The the angles hydraulic of leaves limitation might of also heterophylly reflect the hydraulic in this tree limitation species. 150 Weof heterophylly cannot obtain in the this analytical tree species. solution We cannot of the angleobtainθ thein Equation analytical (3), solution but its numericalof the angle solution θ in Equation can be 151 easily(3), but obtained. its numerical Figure solution8a shows can the be numerical easily obta solutionined. Figure of the leaf8a shows angle the of the numerical population solution of G. of biloba the 152 basedleaf angle on the of Montgomerythe population parameter; of G. biloba Figure based8b on shows the Montgomery the histograms parameter; of the numerical Figure 8b solutions shows the of 153 thehistograms angles of of all the leaves numerical sampled solutions based onof theA/( anglesLW). We of all found leaves that sampled the solution based (= on 128.6 A/(◦LW) based). We onfound the 154 estimatethat the solution of the Montgomery (= 128.6°) based parameter on the is estima approximatelyte of the equalMontgomery to the median parameter of the is estimates approximately of the 155 anglesequal to (= the 129.4 median◦) based of the on allestimates individual of the leaves. angles (= 129.4°) based on all individual leaves.

156 Figure 8. Estimation of the leaf angle of G. biloba.(a) Numerical solution of the leaf angle of the 157 Figure 8. Estimation of the leaf angle of G. biloba. (a) Numerical solution of the leaf angle of the population of G. biloba;(b) histogram of the numerical solutions for different individual leaves sampled. 158 population of G. biloba; (b) histogram of the numerical solutions for different individual leaves In panel (a), the red horizontal straight line represents the estimate of the Montgomery parameter; 159 sampled. In panel (a), the red horizontal straight line represents the estimate of the Montgomery the blue curve represents δ/[4 sin(δ/2)] (see Equation (3)), where δ represents the candidate of leaf 160 parameter; the blue curve represents δ⁄4 sinδ⁄2 (see Equation (3)), where δ represents the angle that was initially set from π/4000 to π by an increment of π/4000; the point at which the red 161 candidate of leaf angle that was initially set from π/4000 to π by an increment of π/4000; the point at straight line and the blue curve intersect is used to quantify the leaf angle of the population (namely all 162 which the red straight line and the blue curve intersect is used to quantify the leaf angle of the the leaves sampled). In panel (b), red bars represent the densities of different groups of leaf angles. 163 population (namely all the leaves sampled). In panel (b), red bars represent the densities of different 164 4.4. Whygroups Do of We leaf Choose angles. to Use the Montgomery Equation? The Montgomery equation shows a simple proportional relationship between leaf area and the 165 4.4. Why Do We Choose to Use the Montgomery Equation? product of leaf length and width. The main value of this parametric equation is that it opens a door to 166 predictThe the Montgomery leaf area based equation on leaf shows length a simple and width proporti amongonal different relationship species between within leaf the area same and taxon. the 167 Inproduct another of relevantleaf length study and [6 width.], we demonstrated The main value that of the this relationship parametric of equation leaf area is vs. that the it product opens a of door leaf 168 lengthto predict and the width leaf forarea closely based relatedon leaf specieslength and followed width theamong Montgomery different species equation. within This the means same that taxon. the 169 dataIn another of leaf arearelevant vs. the study product [6], ofwe leaf demonstrated length and width that the for thoserelationship species withinof leaf thearea taxonomic vs. the product unit such of 170 asleaf a genus,length and a subfamily, width for and closely even related a family species exhibit followed a linear the relationship Montgomery on a equation. planar surface. This means However, that the leaf shapes of closely related species might have a certain degree of variation, which renders it unsuitable to use elliptical Fourier analysis or other more complex/descriptive metrics of leaf shape. Although the above advanced technology could measure the extent of similarity for the leaf shapes, the same leaf shapes might come from species that are unrelated in their genetic relationships. However, the proportional relationship reflected by the Montgomery equation appears to be superior to the Forests 2019, 10, 178 11 of 13 aforementioned complex/descriptive metrics of leaf shape. At least, according to our recent study for six classes of plants, the proportional relationship for the pooled data of leaf area vs. the product of leaf length holds [6]. In this case, we advocate using a parametric model like the Montgomery equation to reveal the relationship among leaf basic morphological indices. We admit that multiple leaf shapes could generate the same or similar estimates of the Montgomery parameter; however, this result (namely the same or similar estimates of the Montgomery parameter for different leaf shapes) might imply that leaves may come from the same ancestor regardless of the difference of leaf shape. For closely related species, the difference in leaf shape does not cause the data of leaf area vs. the product of leaf length and width to largely deviate from a proportional relationship (e.g., the 12 species of Rosaceae reported in ref. [6]). After all, the shoots and leaves of vascular plants experienced an explosive radiation around 430 million years ago [25]. The morphological difference of leaves might result from the distinct gene expression pattern in a particular location like the leaf base [26]. In this case, it is meaningless to use a complex/descriptive metric for measuring the extent of leaf similarity. When we obtain the Montgomery parameter based on a limited sample size (that is surely destructive for sampled leaves), the remaining prediction work is then non-destructive. In the field, it is obviously easier to measure leaf length and width than to measure leaf area. Although there are some non-destructive methods (e.g., Leaf-IT [27]) that could directly measure leaf area in the field, the requirements of the work environment are usually rigorous (e.g., the requirement for a strong color contrast between a leaf and its background). In addition, the Montgomery equation can be simplified to A = cL2 if the coefficient of variation of the ratio of leaf width to length is small [6,8]. As a result, it will further reduce the workload in the field because we only need to measure leaf length for calculating its area. In the present study, we do not state that different leaf shapes must have different Montgomery parameters. In fact, in another study, we also did not say that [6]. We actually intend to show that the estimates of the Montgomery parameters from different species are confined to a small range from 1/2 to π/4. In the present study, we chose four types of leaf shapes that could be described by four explicit mathematical models so that we could accurately derive the theoretical range of the Montgomery parameter corresponding to each leaf shape.

5. Conclusions In the current study, we studied the proportional relationship between leaf area and the product of leaf length and leaf width for four types of leaf shapes (namely elliptical, sectorial, linear, and triangular leaf shapes). We found that the Montgomery equation (A = kLW, where k is the Montgomery parameter) holds for the four types of leaf shapes. By using the elliptical, sectorial, and triangular area formulae, we derived theoretically that the Montgomery parameter should range from 1/2 to π/4; by using the area formula based on the simpliﬁed Gielis equation, we found that the Montgomery parameter for the linear leaf shape ranges from 0.6795 to π/4. We also used the actual leaves of four types of plants to test the above derivations, and we found that the conclusions drawn from the mathematical derivation were supported by the actual data sets. Considering that the estimates of the Montgomery parameter of the six classes of plants we recently reported also fell into the range of (1/2, π/4), we believe that the Montgomery model appears to apply to broad-leaved plants with different leaf shapes. It reﬂects the stability of the proportional relationship between leaf area and the product of leaf length and width during leaf evolution. The Montgomery parameters for different species and taxa were found to be different, which are beneficial for the interpretation of the difference in photosynthetic potentials, because the complexity of leaf shape can have a significant influence on the photosynthetic potentials of plants.

Supplementary Materials: The following are available online at http://www.mdpi.com/1999-4907/10/2/178/s1, Table S1. Collection information of 20 bamboo species; Table S2. The Gielis ﬁt to the leaf edge data of three species of plants. Author Contributions: P.S. and D.A.R. designed the experiment and contributed equally to this work; M.L. and X.Y. carried out the ﬁeld experiment; P.S., J.G., and D.A.R. analyzed the data and wrote the manuscript. All authors read and commented on this manuscript. Forests 2019, 10, 178 12 of 13

Funding: This research was funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (grant number: no number). Acknowledgments: We are thankful to Ping Wang, Xiao Zheng, and Jialu Su for their useful help during the preparation of this manuscript. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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