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2012美国大学生数学建模竞赛 MCM A题


Team #14749

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Summary
Many scholars conclude that leaf shape is highly related with the veins. Based on this theory, we assume the leaf growth in each direction satisfies a function F α . For the leaves in the same tree, the parameters are different; for those of separate trees, the function mode is different. Thus the shape of leaf differs from that of another. In the end of section 3, we simulate one growing period and depict the leaf shape. Through thousands of years of evolution, the leaves find various ways to make a full use of natural resources, including minimizing overlapping individual shadows. In order to find the main factors promoting the evolution of leaves, we analyze the distribution of adjacent leaves and the equilibrium point of photosynthesis and respiration. Besides, we also make a coronary hierarchical model and transmission model of the solar radiation to analyze the influence of the branches. As to the tree structure and the leaf shape, first we consider one species. Different tree shapes have different space which is built up by the branch quality and angle, effect light distribution, ventilation and humidity and concentration of CO2 in the tree crown. These are the factors which affect the leaf shape according to the model in section 1. Here we analyze three typical tree shapes: Small canopy shape, Open center shape and Freedom spindle shape, which can be described by BP network and fractional dimension model. We find that the factors mainly affect the function of ?S that affects the additional leaf area. Factors are assembled in different ways to create different leaf shapes. So that the relationship between leaf shape and tree profile/branching structure is proved. Finally we develop a model to calculate the leaf mass from the basic formula of m = ρ ? V. By adjusting the crown of a tree to a half ellipsoid, we first define the function of related factors, such as the leaf density and the effective ratio of leaf area. Then we develop the model using calculus. With this model, we approximately evaluate the leaf mass of a middle-sized tree is 141kg.

Team #14749

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Dear editor, How much the leaves on a tree weigh is the focus of discussion all the time. Our team study on the theme following the current trend and we find something interesting in the process. The tree itself is component by many major elements. In our findings, we analyze the leaf mass with complicated ones, like leaf shape, tree structure and branch characteristics, which interlace with each other. With the theory that leaf shape is highly related with the veins, we assume the leaf growth in each direction satisfies a function F α . For the leaves in the same tree, the parameters are different; for those of separate trees, the function mode is different. That’s why no leaf shares the same shape. Also, we simulate one growing period and depict the leaf shape. In order to find the main factors promoting the evolution of leaves, we analyze the distribution of adjacent leaves and the equilibrium point of photosynthesis and respiration. Besides, we also make a coronary hierarchical model and transmission model of the solar radiation to analyze the influence of the branches. As to the tree structure and the leaf shape, different tree shapes have different space which is built up by the branch quality and angle, effect light distribution, ventilation and humidity and concentration of CO2 in the tree crown that affect leaf shapes. Here we analyze three typical tree shapes which can be described by BP network and fractional dimension model. We find that the factors mainly affect the function of ?S that affects the additional leaf area. Factors are assembled in different ways to create different leaf shapes. So that the relationship between leaf shape and tree profile or branching structure is proved. Finally we develop the significant model to calculate the leaf mass from the basic formula of m = ρ ? V. By adjusting the crown of a tree to a half ellipsoid, we first define the function of related factors and then we develop the model using calculus. With this model, we approximately evaluate the leaf mass of a middle-sized tree is 141kg. We are greatly appreciated that if you can take our findings into consideration. Thank you very much for your precious time for reading our letter.

Yours sincerely, Team #14749

Team #14749

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Contents
1. Introduction .................................................................................... 4 2. Parameters ...................................................................................... 4 3. Leaves have their own shapes ......................................................... 5 3.1 Photosynthesis is important to plants ...................................... 5 3.2 How leaves grow? .................................................................... 6 3.3 Build our model ....................................................................... 7 3.4 A simulation of the model ........................................................ 9 4. Do the shapes maximize exposure? ............................................... 13 4.1 The optimum solution of reducing overlapping shadows ...... 13 4.1.1 The distribution of adjacent leaves ............................... 13 4.1.2 Equilibrium point of photosynthesis and respiration ... 14 4.2 The influence of the “volume” of a tree and its branches ..... 16 4.2.1 The coronary hierarchical model .................................. 16 4.2.2 Spatial distribution model of canopy leaf area ............. 16 4.2.3 Transmission model of the solar radiation .................... 17 5. Is leaf shape related to tree structure? .......................................... 19 5.1 The experiment for one species ............................................. 19 5.2 Different tree shapes affect the leaf shapes ........................... 21 5.2.1 The light distribution in different shapes...................... 21 5.2.2 Wind speed and humidity in the canopy ....................... 22 5.2.3 The concentration of carbon dioxide ............................ 23 5.3 Conclusion and promotion .................................................... 23 6. Calculus model for leaf mass ........................................................ 24 6.1 How to estimate the leaf mass?.............................................. 24 6.2 A simulation of the model ...................................................... 26 7. Strengths and Weakness ............................................................... 27 7.1 Strengths................................................................................ 27 7.2 Weaknesses ............................................................................ 27 8. Reference....................................................................................... 28

Team #14749

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1. Introduction
How much do the leaves on a tree weigh? Why do leaves have the various shapes that they have? How might one estimate the actual weight of the leaves? How might one classify leaves? We human-beings have never stopped our steps on exploring the natural world. But, as a matter of fact, the answer to those questions is still unresolved. Many scientists continue to study on this area. Recently, Dr. Benjamin Blonder (2010) achieved a new breakthrough on the venation networks and the origin of the leaf economics spectrum. They defined a standardized set of traits – density, distance and loopiness and developed a novel quantitative model that uses these venation traits to model leaf-level physiology. Now, it is commonly thought that there are four key leaf functional traits related to leaf economics: net carbon assimilation rate, life span, leaf mass per area ratio and nitrogen content.

2. Parameters
Sp ?S ∝
P

the area a leaf grows decided by photosynthesis the additional leaf area in one growing period the leaf growing obliquity the total photosynthetic rate the dark respiration rate of leaves the net photosynthetic rate the height of the canopy the distance between two branches the illumination intensity of scattered light from a given direction the solar zenith angle the truck high the crown high

Rd 0

Pn
h
d

id
?
h0
h1

Team #14749

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3. Leaves have their own shapes
3.1 Photosynthesis is important to plants
It is widely accepted that two leaves are different, no matter where they are chosen from; even they are from the very tree. To understand how leaves grow is helpful to answer why leaves have the various shapes that they have. The canopy photosynthesis and respiration are the central parts of most biophysical crop and pasture simulation models. In most models, the acclamatory responses of protein and the environmental conditions, such as light, temperature and CO2 concentration, are concerned[1]. In 1980, Farquhar et al developed a model named FvCB model to describe photosynthesis[2]: The FvCB model predicts the net assimilation rate by choosing the minimum between the Rubisco-limited net photosynthetic rate and the electron transport-limited net photosynthetic rate. Assume An, Ac, Aj μmol CO2 m?2 s ?1 are the symbols for net assimilation rate, the Rubisco-limited net photosynthetic rate and the electron transport-limited net photosynthetic rate respectively, and the function can be described as: An = min Ac , Aj while
cmax i Ac = C +K 1+O i mc

(1)
? Aj = 4C i+8Γ ? R d i ?

V

C ?Γ? K mo

? Rd

J C ?Γ

(2)

where Cj μbar and O mbar are the intercellular partial pressures of CO2 and O2, respectively, K mc μbar and K mo mbar are the Michaelis–Menten coefficients of Rubisco for CO2 and O2, respectively, Γ? μbar is the CO2 compensation point in the absence of R d (day respiration in μmol CO2 m?2 s?1 and J μmol e?m?2 s ?1 is the photosystem II electron transport rate that is used for CO2 fixation and photorespiration[3][4]. We apply the results of this model to build the relationship between the photosynthesis and the area a leaf grows during a period of time. It can be released as: Sp = A n ? S ? T ? K k 1 , k 2 , k 3 , ? , k n (3)

S m2 and T s are the area of the target leaf and the period of time it grows. K k1 , k 2 , k 3 , ? , k n m2 μmol CO2 is a function which can transfer the amount of CO2 into the area the leaf grows and the k1 , k 2 , k 3 , ? , k n are parameters which

Team #14749

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affect Sp. Sp can be used as a constraint condition in our model.

3.2 How leaves grow?
As the collocation of computer hardware and software develops, people can refer to bridging biology, morphogenesis, applied mathematics and computer graphics to simulate living organisms[5], thus how to model leaves is of great challenge. In 2001, Dengler and Kang[6] brought up the thought that leaf shape is highly related to venation patterns. Recently, Runions[7] brought up a method to portray the leaf shape by analyzing venation patterns. Together with the Lindenmayer system (L-system), an advanced venation model can adjust the growth better that it solved the problem occurred in the previous model that the secondary veins are retarded.

Figure 3.1 Terms pertinent to the description of leaf shapes. We know leaves have various shapes. For example, leaves can be classified in to simple leaves which have an undivided blade and compound leaves whose blade is divided into two or more distinct leaflets such as the Fabaceae. As to the shape of a leaf, it may have marginal dentations of the leaf blades or not, and like a palm with various fingers or an elliptical cake. Judd et al defined a set of terms which describe the shape of leaves as follows[8]: We chose entire leaves to produce this model as a simplification. What’s more, they confirmed again that the growth of venations relates with that of the leaf. To disclose this relationship, Relative Elementary Rate of Growth (RERG) can be introduced to depict leaves growth[9]. RERG is defined as the growth rate per distance, in the definitive direction l at a point p of the growing object, yielding RERGl p = ?s
1 d?s dt

(4)

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Considered RERG, the growth patterns of leaves are also different. Roth-Nebelsick et al brought up four styles in their paper[10]:

Figure 3.2 A sample leaf (a) and the results of its: (b) marginal growth, (c) uniform isotropic (isogonic) growth, (d) uniform anisotropic growth, and (e) non-uniform anisotropic growth.

3.3 Build our model
We chose marginal growth to build our model. Amid all above-mentioned studies, we assume that the leaf produce materials it needs to grow by photosynthesis to expand its leaf area from its border and this process is only affected by what we have discussed in the previous section about photosynthesis. The border can be infinitesimally divided into points. Set ∝ as the angle between the x-axis and the straight line connecting the grid origin and one point on the curve, and ?s as the growth distance in the direction of ∝. To simplify the model, we assume that the leaf grows symmetrical. We put half of the leaf into the x-y plane and make the primary vein overlap x-axis.

Figure 3.3 The half of a leaf is settled in x-y plane like this with primary vein overlapping x-axis. The leaf grows in the direction of ∝. This is how we assume the leaf grows. In one circle of leaf growth, anything that photosynthesis provided transfers into the additional leaf area, which can be described as: ?Sn = Sp (5)

Team #14749
xn 0

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x n +1 0

while

yn x dx ?

yn+1 x dx = ?Sn in the figure.

Figure 3.4 The curves of the adjacent growing period and their relationship. In this case, we can simulate leaf growth thus define the leaf shape by using iterative operations the times N a leaf grow in its entire circle①. First, we pre-establish the border shape of a leaf in the x-y plane, yielding y0 x . where x = x0,n ②, the relevant ∝0,1 satisfies: tan ∝0,n =
y0 x 0,n x 0,n y0 x 0,n x 0,n

(6) (7)

∴ α0,n = tan?1

In the first growing period, assume ?s0,n , the growth distance in the direction of ∝0,n , satisfies: ?s0,n = F α0,n


(8)

?s0,n x = ?s0,n ? cos α0,n ∴ ?s 0,n y = ?s0,n ? sin α0,n

(9)

It releases the relationship of the coordinates in the adjacent growing period. In this case, we can use eq.(9) to predict the new border of the leaf after one period of


For instance, if the vegetative circle of a leaf is 20 weeks on average, the times of iterative operations N can be set as 20 when we calculate on a weekly basis. ② The first number in the subscript expresses which period the leaf is in, and the second number represents a typical location on x-axis. Refer to the Figure 3.5. In this case, we assume the beginning state is period 0. Other subscripts occurred in the following context in the same structure are analogical. ③ The function is defined precisely by the nature of the tress. For instance, species A fits well in linear function, species B in power function while that of another kind of tree is exponential.

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growth④: x1,n ′ = x0,n + ?s0,n x And the average simple recursions are⑤: xn+1 = xn + ?sx yn+1 = yn + ?sy (11) y1,n ′ = y1,n + ?s0,n y (10)

Figure 3.5 ?s, ?sx , ?sy and the recurrence relations. After simulate the leaf borders of the interactive periods, use definite integral ⑥ to settle parameters in the eq.(8) then ?s can be calculated in each direction of ∝, thus the exact shape of a leaf in the next period is visible. When the number of times N the leaf grows in its life circle applies above-mentioned recursions to iterate N times and the final leaf shape can be settled. By this model, we can draw conclusions about why leaves have different shapes. For the leaves on the same tree, they share the same method of expansion which can be described as the same type of function as Eq.(8). The reason why they are different, not only in a sense of big or small, is that in each growing period they acquire different amount of materials used to expand its own area. In a word, the parameters in the fixedly formed eq.(8) are different for any individual leaf on the same tree. For the leaves of different tree species, the corresponding forms of eq.(8) are dissimilar. Some are linear, some are logarithmic, some are exponential or mixtures of that, which settle the totally different expansion way of leaf, are related with the veins. On that condition, the characters can be divided by a more general concept such as entire or toothed.

3.4 A simulation of the model
We set the related parameters by ourselves to simulate the shape of a leaf and to
④ ⑤

That means, in the end of period 1. As we both change the x coordinate and the y coordinate, in the new period, these two figures relate through those in the last period in the functions. ⑥ The relationship must meet eq.(5).

Team #14749

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express the model better. First we initialize the leaf shape by simulating the function of a leaf border at the beginning of growing period 1 in the x-y plane. By observation, we assume that the movement of the initial leaf border satisfied: y0 x = x+0.4 ? 0.5
1

(12)

Suppose the curve goes across the origin of coordinates, then the constraint conditions can be: y0 x = x+0.4 ? 0.5 y0 0 = 0
1

(13)

Thus the solution to eq.(13) is: y0 x =
x y 0 0

x dx + y0 0 = ln x + 0.4 ? 0.5x ? ln 0.4

(14)

By using Mathematica we calculate where x′ ≈ 5.3208(cm), y0 x ′ = 0(cm) and the area of the half leaf is: S=
x′ y 0 0

x dx =

x′ ln 0

x + 0.4 ? 0.5x ? ln 0.4 dx ≈ 2.82108 cm2

(15)

Figure 3.6 y0 x = ln x + 0.4 ? 0.5x ? ln 0.4 We settle An = 33 μmol CO2 m?2 s?1 according to the research by S. V. Archontoulisin et al[11] in eq.(3). On a weekly basis, the parameter T = 7 ? 24 ? 60 ? 60 = 604800(s). In eq.(15), we have S ≈ 2.82 cm2 . When assume K k1 , k 2 , k 3 , ? , k n = 8.5 ? 10?4 m2 μmol CO2 ,

Sp = An ? S ? T ? K k1 , k 2 , k 3 , ? , k n = 33 ? 2.82 ? 10?4 ? 604800 ? 8.5 ? 10?4 thus constraint condition eq.(5) becomes: ?Sn = Sp = 4.78 (16)

Team #14749

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When eq.(8) is linear and after referring to Runions’s paper, we assume that Y-value decreases when X-value increases, which means the leaf grows faster at the end of the primary vein. If the grow rate at the end of the primary vein is 0, as y0 0 = π 3, eq.(8) can be described as: ?s1 = b π 3 ? α where b is decided by eq.(16). To settle the value of b, we calculate multiple sets of data by Excel then use a plane curve to trace them and get the approximation of b. In this method, we use grid to approximate ?Sn =
xn 0

(17)

yn x dx ?

x n +1 0

yn+1 x dx.

Apparently, ?Sn (b) is monotone.

Figure 3.7 ?s1 = 2 π 3 ? α When b is 2: The square of each square is 0.0139cm2, and the total number of the squares in the additional area is about 240. So ?S1,2 = 0.0139cm2 ? 240 = 3.33cm2 When b is 2.5:

Figure 3.8 ?s1 = 2.5 π 3 ? α The square of each square is 0.0240cm2, and the total number of the squares in the additional area is about 201. So ?S1,2.5 = 0.0240cm2 ? 201 = 4.82cm2

Team #14749

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When b is 3:

Figure 3.9 ?s1 = 3 π 3 ? α The square of each square is 0.0320cm2, and the total number of the squares in the additional area is about 193. So ?S1,3 = 0.0320cm2 ? 193 = 6.18cm2 After comparing ?S1,2 , ?S1,2.5 , ?S1,3 , we can draw a conclusion that ?S1,2.5 fit eq.(16) best; accordingly, in period 1: ?s1 = 2.5 π 3 ? α (18)

In each growing period, Sp may be different for the amount of material produced is related with various factors, such as the change of relative location and CO2 or O2 concentration, and other reasons. By using the same method, the leaf shape in period 2 or other period can be generated on the basis of the previous growing period until the end of its life circle. What’s more, when eq.(8) is remodeled, the corresponding leaf shape can be changed. With ?s = ?
50x 3 3 25x 2 3

+

? 10x + 1.5, the following figure shows the transformation

of a leaf shape in the first growing period with the same y0 x settled above and we can see that the shape will be dissected in the end.

Figure 3.10 ?s = ?

50x 3 3

+

25x 2 3

? 10x + 1.5 The shape differs from figure 3.7-3.9, as the kind of

function of ?s is different. In this case, it is a cubic model while a linear model in figure 3.7-3.9.

Team #14749

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4. Do the shapes maximize exposure?
4.1 The optimum solution of reducing overlapping shadows
4.1.1 The distribution of adjacent leaves Vein is the foundation of the leaves. With the growing of veins, the leaves gradually expand around. The distribution of main and lateral veins plays an important decisive role in the shapes of leaves. The scientists created a mathematical model which uses three decisive factors - the relationship between the rate of photosynthesis, leaf life, carbon consumption or nitrogen consumption, to simulate the leaves’ shape. Because of carbon consumption is a constant for one tree, and we take the neighboring leaves in the same growth cycle to observe. So, we can only focus on one factor - the rate of photosynthesis. Through the observation of dicotyledon, leaves on a branch will grow in a staggered way that can reduce the overlapping individual shadows of adjacent leaves and make them get more sunlight (Figure 4.1).

Figure 4.1 The rotation distribution of leaves Base on the similar environment, we assume that adjacent leaves nearly have the same shape. From the perspective of looking down, the leaves grow from a point on the branch. So, we can simplify the vertical view of leaves as a circle of which the radius is the length of a vein which is represented with r . The width of the leaf is represented with w . The angle between two leaves is represented with ? (Figure 4.2).

Team #14749

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Figure 4.2 The vertical view of leaves The leaves should use the space as much as possible, and for the leaf with one main vein, oval is the best choice. In general, ? is between 15° and 90° In this way, . effectively reduce the direct overlapping area. According to the analysis of the first question, r and w are determined by the rate of photosynthesis and respiration. Besides, the width of leaf is becoming narrower when the main vein turns to be thinner. 4.1.2 Equilibrium point of photosynthesis and respiration The organism produced by photosynthesis firstly satisfies needs of leaf itself. Then the remaining organism delivered to the root to meet the growth needs of the tree. As we know, respiration needs to consume organism. If the light is not sufficient, organism produced by photosynthesis may no longer be able to afford the materials required for the growth of leaves. There should be an equilibrium point so as to prevent the leaf is behindhand in its circumstances. The formula[12] that describes the photosynthetic rate in response to light intensity with gradual exponential growth index can be expressed as: = ? 1 ?
?
?

(19)

Where P is total photosynthetic rate, Pmax is maximum photosynthetic rate of leaves, a is initial solar energy utilization and I is photosynthetic photon quanta flux density. The respiration rate is affected by temperature,using the formal of index to describe as follows:

Rd ? Rd0 ? 2

T ?25 10

(20)

Where T is temperature and R d 0 is the dark respiration rate of leaves, when the temperature is 25 degrees. As a model parameter, R d 0 can be determined by nonlinear fitting. Therefore, the net photosynthetic rate can be expressed in the index form as follows:

Team #14749
?
? +

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= +

1 ?

?

(21)

Where Pn is the net photosynthetic rate, which does not include the concentration of carbon dioxide and other factors. The unit of Pn is ?mol ? m?2 ? s?1 . We assume that the space for leaf growth is limited, the initial area of the leaf is S0 . At this point, the entire leaf happens to be capable of receiving sunlight. If the leaf continue to grow, some part of the leaf will be in the shadows and the area in the shadows is represented with x . Ignore the fluctuation cycle of photosynthesis and respiration, on average, the duration of photosynthesis is six hours per day. In the meanwhile, the respiration is ongoing all the time. When the area increased to S , we established an equation as follows: ? ? ? 6 ? 3600 ? ? ? 18 ? 3600 = (22) where ? is the remaining organism created by the leaf per day. And from where

? ? 0 , which means the organism produced by photosynthesis has all been broken
down completely in the respiration, we get the equilibrium point.

x?

S ? Pn Pn ? 4 Rd

(23)

The proportion of the shaded area in the total area:

Pn x ? S Pn ? 4Rd

(24)

Cite an example of oak trees, we found the following data (Figure 4.3), which shows the relationship between net photosynthesis and dark respiration during 160 days. The habitat of the trees is affected by the semi-humid monsoon climate.

Figure 4.3 The diurnal rate of net photosynthesis and dark respiration[13] In general, Pn ? 7 ? mol ? m ?2 ? s ?1 , Rd ? 4 ? mol ? m ?2 ? s ?1 . Taking the given numbers into the equation, we can get the result.

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x ? 25 0 0 S From the result, we can see that in the natural growth of leaves, with the weakening of photosynthesis, the leaves will naturally stop growing once they come across the blade between blocked. Therefore, the leaves can always keep overlapping individual shadows about 25%, so as to maximize exposure.

4.2 The influence of the “volume” of a tree and its branches
4.2.1 The coronary hierarchical model Mr. Bōken and Dr. J. Fischer (1987) found that in order to adequate lighting, leaves have different densities and the branches is distributed according to certain rules. They observed tropical plants in Miami, found that the ratio of main branch and two side branches is 1:0.94:0.87, and the angles between them are 24.4° and 36.9° . According to the computer simulation, the two angles can maximize the exposure of leaves. For simply, we use hemisphere to simulate the shape of the canopy. According to the light transmittance rate, we can divide the canopy into outer and inner two layers (Figure 4.3). The volume of the hemisphere depends on the size and distribution of branches.

Figure 4.3 The coronary hierarchical model The angle between the branches will affect the depth of penetration of sun radiation and the distribution of leaves. Besides, thickness of the branches will affect the transfer of nutrients to the leaves. 4.2.2 Spatial distribution model of canopy leaf area Assume that the distribution of leaves is uniform in the section xz but not uniform in the y direction, as the figure 4.3. Then, we can get the formula of leaf area index

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(LAI) as follows[14]:

1 dz ? ?( x, z )dx ? LAI d ? ? d /2 0
? ( x, z ) is the leaf area density function of the micro-body at the point ( x, z ) .

h

d /2

(25)

Where h is the height of the canopy, d is the distance between two branches, and

ax is the function of leaf area which means the distribution of cross-section of the X
direction, and Its value is a dimensionless, defined as follows:
d /2

1 ? ax ( x)dx ? LAI d ? d /2

(26) (27)

ax (x) ? d ? LAI ? Cs (x)

For the canopy which is not uniform in the horizontal direction, the distribution of its leaves is not entirely clear. We use C s to represent the distribution function of the density of leaves. C s can be expressed as a quadratic function or a Gaussian distribution function. 4.2.3 Transmission model of the solar radiation Assume that the attenuation of the solar radiation accords with the law of Beer-Lambert. The attenuation value at a point of canopy where the light arrives with a certain angle of incidence and azimuth is proportional to the length of the path, and the length can be calculated by Goudriaan function. Approximate function of the G function is shown as follows[15]:

G ? G0 ? k (1? 2G0 )cos?
where k is a parameter decided by different plants.

(28)

Take a micro unit in the canopy. Direct sunlight intercepted in this micro unit can be expressed in the following form:

dIb ? ?Ib (z)G(z, ?)dLAI L

(29)

Where Ib means the direct sunlight, ? is the solar zenith angle and dLAI L is the LAI on the path of light.

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Figure 4.4 Micro unit of the canopy Through the integral and the chain rule, we can get the transmittance at the point
( x ', z ') as follows:
h G ( z , ?) ?ax [ x( z )] tan ? sin ?? ? az ( z )? I b ( x ', z ') ? exp(? ? dz ) I b ( h) 1 ? sin 2 ? cos 2 ?? z'

(30)

Average direct transmission rate:

1 tb ( z ') ? ? tb ( x ', z ')dx ' d ? d /2

?

d /2

(31)

Different with the direct light, the scattered light in all directions is intercepted by the leaf surface from the upper hemisphere. The irradiance dId of scattering at the point
( x ', z ') can be expressed as follows:
' dI d ? id ? cos ? ? tb d?

(32)

where id is the illumination intensity of scattered light from a given direction. The transmission rate of the sun scattered light is as follows:

I d ( x ', z ') 1 ? I d ( h) ?
Average scattering transmission rate:
?

2? ? /2

? ?t
0 0
d /2

' b

sin ? cos? d? d?

(33)

1 td ( z ') ? ? td ( x ', z ')dx ' d ? d /2
Global solar radiation reaching the canopy with a given depth z :

(34)

I ( z ') ? I (h)[(1 ? kd ) tb ( z ') ? kd td ( z ')]

?

?

?

(35)

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According to eq.(30) and eq.(33), the maximum depth the solar radiation can reach has a major link with h and ? . The shape of leaves have a relationship with photosynthesis. So, the h and ? of branches do have an influence on the leaves. Results showed that the light level and light utilization of high stem and open center shape as well as small and sparse canopy shape were better than others.Double canopy shape,spindle shape and center shape took second place,while big canopy shape had the lowest light distribution[16].

5. Is leaf shape related to tree structure?
Due to internal and external factors, there are many kinds of tree shapes in the nature. For example, the apple's tree shape is semi-ellipsoidal, the willow's is hemispherical, the peach's likes a cup, the pine's likes cone and so on. The shape of their leaves varies. The leaf shape of apple is oval, pine's is needle, and the Indus's is palm. Is leaf shape related to tree shape? Even for the same species, there are many kinds of tree shapes. For instance, Small canopy shape, Open center shape, Freedom spindle shape, high stem and open center shape, Double canopy shape and so on. The sizes of leaves are different. Does the tree profile/branching structure effects the leaf shape?

5.1 The experiment for one species
To solve this problem, we first consider the relationship of one tree species such as apple, which is semi-ellipsoidal. According to the second question, the hemispherical model is further extended to the semi-ellipsoidal model. Every tree individual shows irregularly because of natural and man-made factors, which reflects on the diversity of the crown in the canopy height direction of the changing relationship between the crown and canopy height. Figure 5.1 shows the relationship.

Figure5.1 Schematic diagram of tree crown contour

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Where h0 means the truck high, h1 means the crown high, d0 , d1,? ? ?, d7 , d8 means crown diameter. Tree shape with the scale change can be characterized in fractal dimension. Across the same scales, a fixed fractal dimension indicates the boundary shape's self-similarity; on different scales, the change of fractal dimension means that different processes or limiting factor has superiority. (Wiens, 1989)[17] According to the application of BP network and fractional dimension, we describe the tree shapes.[18] According to Li Guodong and Zhang Junke's work which detected and evaluated in different tree shapes of ‘Fuji’ apple. We choose three kinds of tree shapes to study; they are Small canopy shape, Open center shape and Freedom spindle shape as figure 5.2 shows.

Figure 5.2 The tree shapes: (a) Small canopy shape, (b) Open center shape, (c) Freedom spindle shape[19] The experiment was taken in an orchard of Fufeng Town, Shaanxi Province, China from July to November in 2006 with 700m altitude. It's flat, north-south trend, sandy loam, middle-management level. The space between two trees is 2.5m ? 3m . The trees have lived 12 years. Select five trees of each shape, divide the crown into 0.5m ? 0.5m ? 0.5m grid. The tree crown was divided into four layers from bottom to top(F1、 F3、 ), with every half meter. From inside to outside, it was divided into F2、 F4 three layers for each half meter. All data are averaged. Table 5.1 shows the data.[20] Table 5.1 Different tree structure
Tree sharps Small canopy shape Open center shape Freedom spindle shape Trunk high/cm 47 70 55 Crown high/cm 260 240 280 Crown diameter/cm 275 275 270 Main branch number outer 3 4 3 middle 3 4 8 inner 2 0 0 Main branch angle 70° 90° 70°

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At the beginning of December, statistic the different tree branch and shoot types in accordance with the crown. If the length is smaller than 5 meter, we suppose short, middle from 5.1m to 15m, rest is long. Table 5.2 shows the data. Table 5.2 The branch quality and species composition of different shapes
layer S F1 F2 F3 F4 Sum 352 141 107 87 687 Small canopy shape M 101 50 28 24 203 L 162 68 35 25 290 Sum 615 259 170 136 1180 S 264 259 234 32 789 Open center shape M 59 51 42 7 159 L 76 59 54 10 199 Sum 399 369 330 49 1147 S 236 340 140 135 851 Freedom spindle shape M 83 135 47 72 337 L 53 81 30 33 197 Sum 372 556 217 240 1385

5.2 Different tree shapes affect the leaf shapes
In each box, use Light quantum meter determination of light intensity, Hygrometer record changes in temperature and humidity, Anemometer with a compass to determine the standard orientation of the anemometer and trees, early, middle, late, each measuring one time, and test the concentration of Carbon dioxide.[21] 5.2.1 The light distribution in different shapes For different tree shapes, the space which is built up by the branch quality and species and the closed level effect Light distribution, ventilation and humidity in the tree crown. Figure 5.3 shows the Light distribution in different shapes.

Figure 5.3 The Light distributions in different shapes. We can see the Light distribution weakens from top to bottom in each shape. The Open center shape has the similar Light distribution in F2, F3 and F4. In F1, the three are similar. In F2 and F3, Freedom spindle shape declines fast. In F4, the Open center shape is higher than the Small canopy shape and Freedom spindle shape. Light levels within the canopy leaf photosynthetic rate determinants of the lack of light will affect

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the blade, thereby affecting the quality of its photosynthetic performance. So that according model 1, it affects the leaf shape. 5.2.2 Wind speed and humidity in the canopy The space which is built up by the branch quality and species and the closed level effect the ventilation and humidity. Increase wind speed, it can speed up the orchard air circulation and the highest concentration of carbon dioxide, which is also good for leaf photosynthesis and flower bud differentiation. It will also increase the temperature difference between day and night within the crown, and reduce crown temperature and relative humidity. Figure 5.4 shows the comparison of wind speed in the canopy of different tree shapes.

Figure 5.4 The comparison of wind speed in the canopy of different tree shapes We can see the Small canopy shape always has the maximum wind speed, then the Open center shape, Freedom spindle shape minimum. In F2, the three shapes have the larger wind speed apparently. Figure 5.5 shows the humidity.

Figure 5.5 The comparison of changing relative humidity in different tree shapes The humidity of the three shapes is almost the same. So we think tree profile or

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branching structure affects the humidity little, which cannot affect the leaf shape here. 5.2.3 The concentration of carbon dioxide From inside to outside, the crown was divided into three layers for each half meter. The space which is built up by the branch quality and species and the closed level effect the concentration of carbon dioxide, directly impact on leaf photosynthesis. Figure 5.6 shows the concentration of carbon dioxide

Figure 5.6 The concentration of Carbon dioxide in different tree shape In Figure 5.4, we know the concentration of Carbon dioxide increases bottom-up. So the leaf photosynthesis increases, which affect the leaf shape. In this figure, we see that Small canopy shape has the maximum density, then the Freedom spindle shape, Open center shape minimum. But the concentration difference is not obvious from inner to outer. So we think it cannot affect the shape from inner to outer.

5.3 Conclusion and promotion
In conclusion, for different tree shapes, the space which is built up by the branch quality and species and the closed level effect Light distribution, ventilation and humidity and concentration of Carbon dioxide in the tree crown. There are the factors which affects the leaf shape. So that the profile/branching structure effects the leaf shape. For the same species, different tree structures, different factors which affect the leaf shape. According to model 1, they effect the An ? min( Ac , A j ) , further effect the
?S n ? S p , that photosynthesis provided transfers into the additional leaf area is

different. As a result, the leaf shape differs. Maybe one factor affects the leaf shape independently. For example, the Light distribution weakens from top to bottom in each shape. Maybe one or more factors work together, for instance, wind speed can speed up the orchard air circulation and the highest concentration of carbon dioxide, which effects photosynthesis. Maybe the factor does not work, for example, the concentration difference is not obvious from inner to outer.

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For the different species, we think different factors influence ?s , it maybe exponential function, Power function, Trigonometric functions, Composite function and so on. The leaf shape of apple is oval, pine's is needle, and the Indus's is palm. But the apple's tree shape is semi-ellipsoidal, the willow's is hemispherical, the peach's likes a cup, the pine's likes cone and so on. With practice, through the survey, we find the leaf shapes related to tree profile/branching structure, leaf index and tree index has a significant negative correlation. leaf index?

tree height Leaf length , treeindex? . Table 5.3 Crowndiameter Leaf width

shows the relationship between leaf shape and tree structure.[22] Table 5.3 The relationship between leaf shape and tree structure. Tree species Hawthorn Apricot Apple Walnut Peach Leaf index 1.0 1.2 1.7 2.5 4.3 Tree shape
Small canopy shape Natural round shape Small canopy shape Open center shape Open center shape

We can see, when the leaf index is 1.0, it probability be Small canopy shape. When the leaf index is 2.5, it may be Open center shape. So that the relationship between leaf shape and
tree profile/branching structure is proved.

6. Calculus model for leaf mass
6.1 How to estimate the leaf mass?
We know that ρ = m V, so m = ρ ? V, which can be used to analyze the leaf mass of a particular tree. In section 5, we have discussed the tree profile. For the type we study most, its horizontal cross-section is a circle with uniform distribution and half ellipse (including circle) serves as its vertical section across centre shaft. As to the lengthwise section, we come up this elliptic curve by setting the primary trunk as the x-axis and the lowest layer as y-axis with x-axis being symmetry axis of this ellipse. The function can be described as
x2 a2

+ b2 = 1 x ≥ 0 ;

y2

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b2 b2

page 25 of 29

So

y x =

1 ? a 2 x 2 and y x = ? 1 ? a 2 x 2

x≥0 .

Figure 6.1 The elliptic curve simulating the tree crown. Take calculus in to consideration. When we fractionize x-axis into infinite parts that each part can be regarded as a dot, the y-value calculated by the x-value of the dot is the radius of a circle which represents the horizontal cross-section at the height of the x-value of the vary tree. The volume of the tree is the integral of the area of each circle within the whole height of the tree. According to section 4, leaves are different considering the location in the branch. We suppose in the each y-axis direction in the Figure 6.1, the density of leaf satisfies a function related with the location, that is, ρ y . It is affected by various factors. For instance, it is possible that at the point farther from the bottom of the branch, leaves start to germinate later and receive more sunshine thus its photosynthesis rate is faster[23] and the leaves store more materials to process photosynthesis, which make the leaf density higher. On each height of the circle plane, because of the various shape of the leaves and their distributions, the entire round area is not fully occupied by leaves, thus the number of dividing the circle area by the leaf area is less than one. We defined the ratio on different levels within the tree height as φ x expressed as the effective ratio of leaf area. We know that the tree profile is related with the number, thickness and angle of the branches which will have an effect on the circumstances, such as the concentration of CO2 and O2, humidity and wind speed. Those conditions will influence the leaf shape. The leaf shape coupled with the characteristic of branches determines that all the leaves cannot fill out the space of a tree volume, so we use φ x to describe this relationship. Generally speaking, φ x is very small by our obversation. Assume at the same plane, the leaf density satisfies the same distribution as ρx y ,
0

with the height of ?x → 0, the mass of this volume can be described as:

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?m

x=x 0

= φ x0

y x0 0

2πr ? ρx r dr ? ?x
0

?x → 0

(36)

where the subscript x0 in ρx r represents the height level of the leaf layer. Each
0

layer may have its own expression of leaf density so the ρ r may be different. Within the height a, we divide it into n parts on average with each part equals to a/n. By the basic concept of calculus, we can develop the model by integral for two times: m=
a 0

?m dx = limn→∞

a n t=1 φ n

?t ?

y ?t n 0

a

2πr ? ρa ?t r dr
n

(37)

6.2 A simulation of the model

Figure 6.2 The elliptic curve with definite parameters. We assume the data in this model to predict the mass of the leaves with the height (with leaves) of 400cm and the radius of the circle at the bottom to be 180cm. The lengthwise section is still simulated as half of a ellipse. The function is 1 x≥0 Suppose thatρ r is fixed in a vertical line, so eq. (37) can be simplified as: m=
400 0 180 2 ? 0
81 2 x 400

+ 180 2 = 400 2

x2

y2

2πr ? ρ r dr ? φ x dx

(38) g?

Considering what have been mentioned, we assume ρ r = 0.87 ? 7.7 ? 10?4 r cm?2 .[24]

Other ideas conclude that leaves on the top grow sparser to allow the lower leaves expose to more sunshine, thus we settle φ x = 0.1 ? 2.25 ? 10?4 x

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Then:
400 81 2 180 2 ? x 400 0

m=
0

2πr ? 0.87 ? 7.7 ? 10?4 r dr ? 0.5 ? 2.25 ? 10?4 x dx

= 141059 g = 141.059 kg It means for a tree with an ellipsoidal crown which height is about 4m and maximum radius is about 1.8m, the leaf mass is approximately 141kg.

7. Strengths and Weakness
7.1 Strengths
Our model analyzes the tree shape from principle. First we consider one piece of leaf that how its shape changes during grown process. Second we think about one tree about how illumination and tree branches affect the shape. Finally we consider trees of different species, and the relationship between tree structure and leaf shape. Our study increases by degrees. Innovatively, we chose marginal growth to build our model. The leaf growing is related to the direction, and the product that photosynthesis provided transfers into the additional leaf area. To change the function, we get a variety of tree shapes. We search for a number of information, and improve the model on the basis of the previous work. What’s more, we quote the key factors of light distribution, ventilation and humidity and the concentration of carbon dioxide. We can get the results from our model. The relationship between tree structure and leaf shape is described by diagrams and specific values, which is intuitive and vivid.

7.2 Weaknesses
Our second model is not simple enough due to the complexity of nature. We don't simplify enough the key factors of light distribution, ventilation and humidity and the concentration of carbon dioxide. It’s better for us to make it more abstract and clearer. We have difficulties in finding actual data of cases, and fit the data to our model. By solving the problem, our models may be better and more realistic for application. Also, sensitivity and stability test is not conducted, which we should take to test our conclusion.

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8. Reference
[1] Ian R. Johnson, John H. M. Thornley,et al. A model of canopy photosynthesis incorporating protein distribution through the canopy and its acclimation to light, temperature and CO2. Annals of Botany, 2010, 106: 735–749. [2] Farquhar GD, von Caemmerer S, Berry JA. A biochemical model of photosynthetic CO2 assimilation in leaves of C3 species.Planta, 1980, 149: 78–90. [3] Farquhar GD, von Caemmerer S. Modelling of photosynthetic response to environmental conditions. In: Lange OL, Nobel PS,Osmond CB, Ziegler H, eds. Physiological plant ecology II. Water relations and carbon assimilation. Encyclopaedia of plant physiology, New Series, 1982,Vol. 12 B. Berlin: Springer Verlag: 549–588. [4] Yin X, van Oijen M, Schapendonk AHCM. Extension of a biochemical model for the generalized stoichiometry of electron transport limited C3 photosynthesis. Plant, Cell and Environment 2004,27: 1211–1222. [5] Prusinkiewicz, P. Visual models of morphogenesis. Artificial Life, 1994, 1, 1/2:61–74. [6] Dengler N,Kang J.Vascular patterning and leaf shape[J].Current Opinion in Plant Biology,2001,4:50-56. [7] Runions A,Fuhrer M,Lane B,et al.Modeling and visualization of leaf venation patterns[J].ACM Transactions on Graphics, 2005, 24(3):702-711. [8] Judd W W,Campbell C S,Kellogg E A,et al.Plant systematic : A phylogenetic approach[M].Sunderland,MA:Sinauer Associates,1999. [9] Hejnowicz, Z., Romberger, J. Growth tensor of plant organs. Journal of Theoretical Biology, 1984, 110: 93–114. [10] Roth-nebelsick, A., Uhl, D., Mosbugger, V., Kerp, H. Evolution and function of leaf venation architecture: a review. Annals of Botany, 2001, 87: 553–566. [11] S. V. Archontoulis, X. Yin,et al. Leaf photosynthesis and respiration of three bioenergy crops in relation to temperature and leaf nitrogen: how conserved are biochemical model parameters among crop species?. Journal of Experimental Botany, 2012, 63, (2):895–911. [12] Penning de Vries F.W.T., Van Laar, H.H. and Chardon, Potential Productivity of Field Crops under Different Environments, 1983, International Rice Research Institute, Los Banos, Philippines, pp.37-59. [13] Sun Shucun, Chen Lingzhi, Leaf Growth and Photosynthesis of Quercus Liaotungensis in Dongling Mountain region, [J], Aacta Ecologica Sinica, 2000.3.

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[14] Shi Weimin, Chen Qingyun, Constructing of Cucumber Canopy Photosynthesis Model in Sunlight Greenhouse, [J], Journal of Huazhong Agricultural University, 2004.12, 134~139. [15] Yang X, H TED, SHORT. Plant Architectural Parameters of A Greenhouse Cucumber Row Crop, Agricultural and Forest Meteorology, 1990, 51: 93~105. [16] Li Enmao, Yang Yang, Chen Feng, Zhai Heng, The Light Distribution of Different Tree Shapes and Light, [J], Chinese Agricultural Science Bulletin, 2008.10. [17] Wiens A, Spatial scaling in ecology. Functional Ecology, 1989,3:385-397. [18] Zhao Maocheng,Research on Tree Crown Recognition System Based on BP Networks, SCIENTIA SILVAE SINICAE 2004,40(1). [19] Guo Yongcheng, Suitable for pear cultivation of high yield tree, Jilin Animal Husbandry and Veterinary, 1998,01. [20] LI Guodong, Zhang Junke,Comparative Study of the Ecological Factors in Different Tree Canopy Shapes for‘Fuji’Apple, JOURNAL OF NORTHWEST FORESTRY UNIVERSITY, 2008,23(1). [21] Li Enmao,The Light Distribution of Different Tree Shapes and Light Utilization in 'Fuji' Apple Tree, CHINESE AGRICULTURAL SCIENCE BULLETIN, 2008,24(10). [22] Lv Zhengtao, An interesting observation between leaf shape and tree shape, Deciduous fruit trees, 1986,04. [23] Li Enmao, Yang Yang, et al. The Light Distribution of Different Tree Shapes and Light Utilization in ‘Fuji’ Apple Tree. Chinese Agricultural Science Bulletin. 2008, 24,(10):347-350. [24] Jiang De-an, Li Guo-yuan, et al. Correlation and Selectivity of Leaf-used Ginkgo Vaviety Leaf Form lndicator and Leaf Output. Journal of Fujian College of Forestry. 2004, 24, (2):169-171.


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