2012美国大学生数学建模竞赛a题获奖论文

1、Classify Leaf Shape and Estimate for Leaf MassWe build a mathematical model to classify leaves and calculate the weight of leaves of a tree.First, we consider the reasons for forming various leaf shapes. Our answer is that the various leaf shapes are response to the various environmental factors suc

2、h as carbon dioxide, temperature and water. After investigating the vertical distribution and the level distribution of natural zones in Alps and China, we make the conclusion that the size of shape is in negative relation with the carbon dioxide level, while positive with temperature and the amount

3、 of water.Next, in order to figure out whether the shapes maximize exposure, we analyze leaf mosaic and get the answer yes. And by comparing the surface leaves and the leaves below, we know that the distribution of leaves do influence leaf shape since the surface leaves are generally smaller and hav

4、e more complex edges in order to let the sunlight penetrate to the leaves below, while the leaves below are larger and have relatively simpler margin .Then, to analyze the relationship between leaf shape (LS) and tree profile (TP), we build a simple linear regression model. First, we assume that the

5、re are 5 kinds of LS and 8 kinds of TP and transfer them to numbers. Then we investigate 120 kinds of trees randomly selected from a botany encyclopedia and classify their LS and TP by labeling numbers. From the scatter, we find that the two variables are positively correlated. Thus we build a simpl

6、e linear regression model and get the linear regression equation y = 1.3567x + 0.6536 ( y stands forTP, x stands for LS). And since the regression coefficient is 1.3567, we can come to the conclusion that leaf shape is, to a large extent, similar to the general profile of tree. Finally we make a sig

7、nificant level test and get the correlation coefficient is 0.748993, indicating that the positive correlation between LS and TP is highly significant.Finally, in order to calculate the weight of the leaves, we define a normal tree whose outline crown is relative axis-symmetric, and then we shrink th

8、e outline of the crown to form a serials of crowns: the leaves distribution and their characteristics in each spatial defined as a section between two adjacent crowns is similar. By sampling from the section, we can macro-analyze the overall condition of leaves; calculate the leaves mass in the sect

9、ion and then add up all the sections weight to get the result. In addition, we also try to modify this model by applying calculus to the calculation of complicated shape tree via dividing the tree into several axis-symmetric little trees as sub-models.In a word, the strong theoretical basis and suit

10、able assumption make our model estimable for further study of leaves. Combining this model with more accurate information of the tree shape and leaves density will maximize its utility.Team # 13828Page 1 of 22AbstractWe build a mathematical model to classify leaves and calculate the weight of leaves

11、.First, we consider the reasons for leaves varying hape and size and focus on the factors including carbon dioxide, temperature and water. Combined with previous researches, we analyze the vertical distribution and the level distribution of natural zones in Alps and China and get the result that as

12、the temperature decreases and the amount of water declines, the leaf shape is inclined to be needle-shaped and as the drop of carbon dioxide level, the leaftend to be bigger.Then, we analyze leaf mosaic and know that leaves tend to shape themselves to maximize the exposure to sunlight. Whats more, w

13、e draw a conclusion that leaves locating at different parts of a tree often have different shapes since the surface leaves are generally smaller and have more complex edges or lobes while the leaves below are much larger and the leaf margin and lobes arerelatively simpler.Next we build a simple line

14、ar regression model to analyze the relation between leaf shape and tree profile, from which we get the linear regression equation. Besides, we make a significance test and get that correlation coefficient is 0.748993, indicating that the model is of high significance. According to the regression mod

15、el, we know that there is positive correlation between leaf shape and tree profile: leaf shape is, to a large extent, similar to the general profile oftree.Finally, as for the weight of leaves, we divide the total tree into several parts which share the same characteristics. And we work out the leaf

16、 mass of eachsection and then add up all the sections to estimate the total leaf mass of a tree.Furthermore, combining this model with more accurate information of the tree shape and leaves density will maximize its utility.Key word: Leaf Shape; Leaf mass; Simple linear regression; PartitionTeam # 1

17、3828Page 2 of 22ContentsAbstract1Introduction3Restatement and analysis of the problem4The Basic Assumption5The Tree is Unchangeable5Number the Shape5The Tree is Normal5Model6Question16Question28Question38Question4114.1 Symbols and Definitions124.2 Basic Model124.3 Improved Model I154.4 Improve Model

18、 II16Simulation and Analysis16Strengths and Weaknesses18Summary and Recommendations18A letter to the science journal19References20Attachment21Team # 13828Page 3 of 22IntroductionWe build a mathematical model to classify leaves and calculate the weight of leaves.First, we consider Why do leaves have

19、different shapes. And our answer is that the various shapes of leaf are response to the various environments. To analyze the relation between leaf shape and environment, we focus on factors including carbon dioxide, temperature and water. About carbon dioxide, combining with previous researches, we

20、make a conclusion that with diminishing concentration of carbon dioxide, the leaf tends to be bigger. As for temperature and water, we respectively investigate the vertical distribution and the level distribution of natural zones in Alps and China and get the result that as the temperature decreases

21、 and the amount of water declines, the leaf shape isinclined to be needle-shaped.Next, in order to figure out whether the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure we analyze a phenomenon called leaf mosaic: the distribution of leaves in a single plan

22、e is usually perpendicular to light rays, which provides the least shading of leaves by one another (Great Soviet Encyclopedia). And by comparing the surface leaves and the leaves below, we know that the surface leaves facing plenty of sunlight, are generally smaller, so that they can reduce the sur

23、face area to absorb light and make a lot of sunlight penetrate to the leaves below, which is good for the rest of leaves to carry out photosynthesis. Whats more, canopy leaves generally have complex edges or lobes, which makes the leaves quickly scattered the heat absorbed. The leaves below are easy

24、 to be obscured so they are generally larger than the canopy leaves for absorbing more light and theleaf margin and lobes are relatively simpler.Then, to analyze the relationship between leaf shape and tree profile, we build a simple linear regression model. First, we assume that there are 5 kinds o

25、f leaf shape and 8 kinds of tree profile and transfer them to numbers. Then we investigate 120 kinds of trees which are randomly selected from the menu of a botany encyclopedia written by Christopher Brickell and classify these leaf shape and tree profile by labeling numbers. From the scatter we dra

26、w with excel, we find that the two variables are positively correlated. Thus we build a simple linear regression model and get the linear regression equationy = 1.3567x + 0.6536 ( y stands for tree profile, x stands for leaf shape). AndTeam # 13828Page 4 of 22since the regression coefficient is 1.35

27、67, we can get the result that leaf shape is, to a large extent, similar to the general profile of tree. Finally we make a significant level test of parameters and the equation, and get the correlation coefficient is 0.748993, indicating that the positive correlation between leafshape and tree profi

28、le is highly significant.Finally, in order to calculate the weight of the leaves, we define a normal tree whose outline of crown is relative axis-symmetric, and then we shrink the outline of the crown to form a serials of crowns; the leaves distribution and characteristics in each spatial defined as

29、 a section between two adjacent crowns is similar. By sampling from the section, we can macro-analyze the overall condition of leaves, calculate the leaves mass in the section, and then add up all the sections weight to generate the result. We also try to modify this model by applying calculus to th

30、e calculation of complicated shape tree viadividing the tree into several axis-symmetric little trees as sub-models.In a word, we build a model to solve the problem of leaves. The strong theoretical basis and suitable assumption make it estimable for further study of leaves. Combining this model wit

31、h more accurate information of the tree shapeand leaves density will maximize its utility.Restatement and analysis of the problemThe problem needs us to build a mathematical model to describe and classify leaves. Consider and answer the following:The reason why leaves have the various shapes.Whether

32、 the shapes “minimize” overlapping individual shadows, so as to maximize exposure and what the correlation between the distribution of leaves and leaf shape.The correlation between leaf shape (general characteristics) and tree profile/branching structure.Estimate of leaf mass of a tree and analyze t

33、he correlation between the leaf mass and the size characteristics of the tree (height, mass, volume definedby the profile).We analyze the problems above and then come up with the following approach towards result. About question 1&2, we need to study previous results and analyze someexamples to get

34、answers.Team # 13828Page 5 of 22 As for question 3, we need to classify leaf shape and tree profilebylabeling numbers and build a regression model to find the correlation between them. To solve question 4, we need to come up with a model to estimate leaf mass. And in light of the correlation among t

35、he leaf mass, the size characteristics of the tree and the distribution of leaves, we try to usedivision thought to solve the problem.The Basic AssumptionThe Tree is UnchangeableLeaf shape and tree profile are relatively fixed and we only consider the profile in the mature period. A few kinds of tre

36、es crown profile change through lifetime, but these only take up a small percentage of all the tree species. Therefore, we ignore them and only take the unchangeable tree into consideration in this model.Number the ShapeThe leaves have various shapes such as elliptic (oval, with a short or no point)

37、, digitate (divided into finger-like lobes), lanceolate (long, wider in the middle), scale shape and needle shape. We number the different shapes from 5 to 1, whose shapes are from broad to needle, and we make it easy to describe andclassify them.Similarly, there are 8 kinds of tree profiles, which

38、we classify by crown shape: globular, spheroid, umbrella, egg, verticillate, tower, wide conic, conic, which are numbered from 8 to 1. These 8 kinds almost contain all kinds of crown shape.The Tree is NormalWe use the word normal to denote the features of a tree that simplify our treatment: The enti

39、re tree is healthy, mature and naturally grown up with little sufferings from the human and animals or other natural disasters such as lightning, typhoon, and tornado. To better understand this, the analysis of a tree should have the following characteristics. The outline of the crown is generally a

40、xial symmetric to the center of trunk. We assume that the outline shape of a tree is basically in the center of the axis for trunk for axisymmetric, which will not differ with thechange of species.Team # 13828Page 6 of 22 The leafs distribution is constrained. We presume the leaf density of the tree

41、 (the number of leaves per unit volume) will increase gradually from the center of the trunk to the margin of the crown. The leafs size is in negative-relationship with the leaf density. We hypothesize that the greater the leaf density, the smaller the size of leaf is; from the center of the trunk t

42、o the crown, the leaves distribution density isgradually increasing while the size of the leaves is gradually decreasing.ModelQuestion1Why do leaves have different shapes?There are lots of reasons for leaves varyinghape and size. All in all, thevarious shapes of leaf are response to the various envi

43、ronments. For example, in desert, trees shape their leaves like needles to keep water from evaporation while in tropical rain forest, trees have broad leaves to let go of the excess water. In the following part, we consider the relation between leaf shape andenvironmental factors including carbon di

44、oxide, temperature and water.About carbon dioxide, Benjamin Blonder came up with an idea that it is among the three main factors (the amount of carbon required to make it, how long theleaf lives and how fast or slow it processes sunlight) that create an incrediblediversity of leaf shapes and structu

45、res. In addition,ina Science, the authorpoints out, according to the PNAS(Proceedings of the National Academy of Sciences of the United States of America) in 2004, British scientists have pointed out that the reason why primitive land plants in ancient times later evolved to large leaves to absorb s

46、unlight is the continuous decrease in concentration of carbon dioxide. Besides, David Beerling and Colin Osborne inthe University of Sheffield, along with William Chaloner in the University ofLondon, put forward a theory that at that time, the extremelyhighconcentration of carbon dioxide impeded the

47、 leafs process of evolution. They also pointed out that, the concentration of carbon dioxide in Devonian atmospheric is 10 times higher than that now and the stomas of leave which are used to let carbon dioxide in are much less. Thus, we can conclude that there is a negative correlation between leaf

48、 size and concentration of carbon dioxide: asthe carbon dioxide level decreases, the leaf becomes larger.Team # 13828Page 7 of 22As for temperature and water, we respectively investigate the verticaldistribution and the level distribution of natural zones in the Alps and China.Figure 1 Vertical dist

49、ribution of natural zones in AlpsFrom the vertical distribution of natural zones in Alps, we can conclude that as the altitude rises and temperature decreases, natural zones vary from deciduous broad-leaved forest to coniferous forest and leaf shapes vary from broadness tonarrow.Figure 2 the level d

50、istribution of natural zones in ChinaFrom Figure2 above, we can find that from seaside to inland, as the amount of water declines, there are tropical rain forest, broadleaf forest, coniferous and broad-leaved forest, coniferous forest under Alt.1000m where leaf shape varies from broadleaf to needle-

51、shaped leaf. Thus, we can conclude that as thetemperature decreases and the amount of water declines, the leaf shape isTeam # 13828Page 8 of 22inclined to be needle-shaped.Question2Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distributio

52、n of leaves within the “volume” of the tree and its branches effect the shape?The phenomenon that leaves try to minimize overlapping individual shadows to maximize sun exposure is common among trees, which is called leaf mosaic in biology. Leaf mosaic is the distribution of leaves of plants in a sin

53、gle plane, usually perpendicular to light rays, which provides the least shading of leavesby one another (Great Soviet Encyclopedia).In another word, some leaves changes their shapes in order to maximize the sunlight received. Besides, Leaf mosaic is caused by the unequal growth of petioles and leaf

54、 blades that are drawn to light and fill every space exposed tothe suns rays. As a result, the size and even the shape of leaves are altered.As for distribution, leaves locating at different parts of a tree often have different shapes. Facing plenty of sunlight, the surface leaves are generally smal

55、ler, so that they can reduce the surface area to absorb light and make a lot of sunlight penetrate to the leaves below, which is good for the following leaves to carry out photosynthesis. Whats more, canopy leaves generally have complex edges or lobes, which makes the leaves quickly scattered the he

56、at absorbed. The leaves below are easy to be obscured so they are generally larger than the canopy leaves for absorbing more light. And the leaf margin and lobesare relatively simpler.Thus we can conclude that the shapes do “minimize” overlapping individual shadows that are cast, so as to maximize e

57、xposure and the distribution of leaves within the “volume” of the tree and its branches does affect the shape.Question3Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure?To demonstrate this question, this paper builds a simple linear regression model to analyze the relation between leaf shape and tree profile. Step1: Sore the ShapeTeam # 13828Page 9 of 22We investigated 120 kinds of trees which are selected from the menu of Encyclopedia of plants a