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A STUDY OF THE LENGTHS AND WIDTHS

OF OAK TREE LEAVES

by

THE MIGHTY OAKS

Edwin Baumgartner
Timothy Miller
Gayle Smith
David Truscott

Study of the lengths and widths of oak tree leaves.

Introduction:

The group studied the (anticipated) relationship between the width and length of leaves, for two varieties of oak trees. It was expected that the ratio of width to length would be constant for a particular variety.

For one tree, data was categorized by the number of leaves in each cluster. The question arose as to the amount of variation among the clusters and the possible relationship of this to the "distance" between clusters.

Description of data, collection methods, units of measurement:

Two oak trees, one a red oak and the other an unidentified variety of oak, on the Dickinson College campus, were chosen. From the red oak a "random" sample of leaves from reachable branches was selected. A wind-felled branch from the second oak constituted the other sample.

For each leaf in the sample, we recorded the length and width in millimeters. The length was measured along the stem vein from the tip to the base of the leaf (fleshy part of leaf), without consideration to curvature. The width was defined to be the widest distance between the tips of the lobes of a leaf running roughly perpendicular to the vein.

The collected data separates measurements by tree and by cluster of leaves. Cluster data was only collected from the unidentified variety of oak. Due to time restraints, the cluster information was not incorporated in the following report.

Link to the dataset

Descriptive Statistics:

The following box plot demonstrates the variation within the samples for each of the trees and illustrates that the range of lengths appear to be different for the two trees.

Box plot of length

The following scatterplot indicates a possible linear relationship betwen width and length. It is apparent that the relationship is different for the two trees.

A summary of the basic statistics follow:

     Tree #  Sample Size  Mean Median

length   1        42      13.5  13. 
         2        48      17.6  17.4

width    1        42      12.3  12.8
         2        48      13.0  12.5

ratio    1        42      0.91  0.92
         2        48      0.73  0.73

The Minitab display of the complete descriptive statistics are contained in the appendix.

Next , consider the regression information for each tree, separately.

For the red oak ( in scatterplot as tree #1, symbol black o)

The regression equation is width = 7.75E -02 + 0.904 length. The constant term indicates that the line passes through the origin as would be expected. The linear coefficient shows that the expected width is 0.904 times the length.The corresponding residual plot seems to show no noticable pattern, meaning that the linear regression line provides a satisfactory model.

For the second oak tree (scatter plot Tree#2, symbol +).

The regression line for tree #2 is width = -1.21 + 0.803 length. The linear coefficient shows that the expected width is 0.803 times the length. The corresponding residual plot seems to show no noticable pattern, indicating that the linear regression line provides a satisfactory model.

Results of the pilot study:

It appears that there is a strong relationship between length and width of leaves for the two oak trees and that this ratio was unique for the two varieties.

Unanswered questions and recommendations for future study:

  1. A complete study should start with the active participation of a subject matter specialist - a tree expert. Associated with this should be a more extensive literature search of known information on this topic.

    A large concern is the clearly non-random sampling methods used in this pilot study. A more complete study would consider leaf location - top vs. bottom, interior leaves vs. exterior leaves, sun vs shade. Also, trees from different geographical locations could be chosen -- rural vs. urban, open space vs. dense forest. The possible effect of tree age, tree health on the ratio of length to width could also be studied. This list is not exhaustive.

  2. For the unidentified variety of oak tree, it was noted that the leaves grew in clusters of from 4 to 8 leaves. Another study might assess the relationship of leaf dimensions to the number of leaves in a cluster, and the spatial relationship among clusters.

  3. It would be interesting to follow the growth patterns of specific leaves over a period of time from first emergence in spring to full growth in mid-summer.

  4. Finally, if the ratio of length to width is unique to each variety of oak , then that ratio can aid in the identification of the variety of an oak tree.


APPENDIX:

Comparison of leaf dimensions between trees.

To study the relationship between the dimensions of the leaves a scatter plot and related regreeeion analysis was performed for the width (y) vs. length (x) for each of the trees.

The regression equation for tree 1 is: wid.1 = 0.077 + 0.904 len.1

The regression equation for tree 2 is: wid.2 = - 1.22 + 0.803 len.2

Regression Analysis for tree 1:

The regression equation is
wid.1 = 0.077 + 0.904 len.1

Predictor       Coef       Stdev    t-ratio        p
Constant      0.0775      0.8770       0.09    0.930
len.1        0.90444     0.06389      14.16    0.000

s = 1.045       R-sq = 83.4%     R-sq(adj) = 82.9%

Analysis of Variance

SOURCE       DF          SS          MS         F              p
Regression    1      218.74      218.74    200.40    0.000
Error              40       43.66        1.09
Total              41      262.40

Regression Analysis for tree 2:

The regression equation is
wid.2 = - 1.22 + 0.803 len.2

48 cases used 2 cases contain missing values

Predictor       Coef       Stdev    t-ratio        p
Constant     -1.2156      0.8627      -1.41    0.166
len.2        0.80264     0.04817      16.66    0.000

s = 1.122       R-sq = 85.8%     R-sq(adj) = 85.5%

Analysis of Variance

SOURCE       DF          SS          MS         F        p
Regression    1      349.29      349.29    277.62    0.000
Error        46       57.87        1.26
Total        47      407.16

Variable     TREE.NO      N   NMiss   Mean    Median   TrMean
length           1       42      0   13.493   13.600   13.558
                 2       48      2   17.592   17.400   17.616

width            1       42      0   12.281   12.800   12.326
                 2       50      0   12.950   12.450   12.925

ratio            1       42      0   0.9103   0.9239   0.9140
                 2       48      2   0.73159  0.73186  0.73273

Variable   TREE.NO    StDev   SEMean      Min      Max       Q1       Q3
length           1    2.554    0.394    7.900   17.600   11.550   15.925
                 2    3.396    0.490    9.800   25.200   15.175   19.775
width            1    2.530    0.390    7.000   16.600   10.325   14.200
                 2    2.908    0.411    7.100   19.300   10.650   15.450
ratio            1   0.0773   0.0119   0.6952   1.0462   0.8738   0.9553
                 2  0.06533  0.00943  0.57303  0.86624  0.68178  0.78643

DATA: Tree #1

length width stem lobe ratio cluster tree TREE.NO
 14.0   13.3  3.1   7 0.95000   *      1
 15.2   11.0  4.3   9 0.72368          1
 16.0   15.6  5.4   7 0.97500          1
 14.7   13.8  4.9   7 0.93878          1
 11.1   11.5  2.5   7 1.03604          1
 10.1   10.1  2.3   7 1.00000          1
  8.8    8.8  2.4   7 1.00000          1
  7.9    7.0  1.3   7 0.88608          1
 14.3   13.1  4.5   7 0.91608          1
 16.7   14.2  5.6   9 0.85030          1
 16.3   12.7  5.1   7 0.77914          1
 13.4   13.7  3.8   7 1.02239          1
 17.2   16.0  6.5   7 0.93023          1
 12.2   11.3  2.4   7 0.92623          1
 10.0    8.6  2.4   7 0.86000          1
 13.6   13.0  4.2   7 0.95588          1
 10.4    8.6  2.2   7 0.82692          1
 13.6   11.1  4.2   7 0.81618          1
 12.8   12.2  3.5   7 0.95312          1
 13.1   11.6  4.5   7 0.88550          1
 13.0   13.6  4.3   7 1.04615          1
 14.1   13.7  4.3   7 0.97163          1
 15.9   14.2  5.3   7 0.89308          1
 12.1    9.9  3.1   7 0.81818          1
 12.5   10.4  3.4   7 0.83200          1
 16.0   14.3  3.7   7 0.89375          1
 12.3   11.4  2.9   7 0.92683          1
 14.8   13.0  3.1   7 0.87838          1
 16.6   16.6  3.4   7 1.00000          1
 10.5    7.3  1.7   9 0.69524          1
 10.2    9.3  1.9   7 0.91176          1
 13.6   12.6  4.8   7 0.92647          1
 15.6   14.6  4.6   7 0.93590          1
 17.6   16.5  5.1   7 0.93750          1
 16.0   15.9  5.2   7 0.99375          1
 16.7   15.1  4.3   7 0.90419          1
 15.6   14.9  4.9   7 0.95513          1
 10.2    9.4  2.9   7 0.92157          1
  9.6    8.7  1.9   7 0.90625          1
 14.2   12.9  3.0   7 0.90845          1
 11.7   10.9  2.8   7 0.93162          1
 16.5   13.4  4.1   9 0.81212          1

DATA: Tree #2

length width stem lobe ratio  cluster tree
   *    12.5  5.0      *         G      2
 12.4    9.8  3.5     0.79032    G      2
 19.8   15.1  5.6     0.76263    G      2
 15.7   11.0  4.8     0.70064    G      2
 21.2   17.3  6.0     0.81604    G      2
 17.1   13.0  5.0     0.76023    G      2
 15.7   11.4  3.7     0.72611    E      2
 19.2   14.1  9.1     0.73438    E      2
 17.4   12.8  4.0     0.73563    E      2
 16.7   13.6  4.7     0.81437    E      2
 23.3   16.8  5.2     0.72103    E      2
 20.6   16.8  5.5     0.81553    E      2
 17.7   11.5  5.0     0.64972    E      2
 18.8   12.1  3.9     0.64362    F      2
 18.7   14.8  3.7     0.79144    F      2
 14.7    9.9  3.6     0.67347    F      2
 25.2   19.3  6.1     0.76587    D      2
 23.0   17.0  5.5     0.73913    D      2
 16.5   13.9  5.3     0.84242    D      2
 20.8   16.5  4.8     0.79327    D      2
 18.2   11.7  6.2     0.64286    D      2
 16.0   10.2  4.1     0.63750    D      2
 23.4   17.3  4.9     0.73932    D      2
 14.8    8.8  3.9     0.59459    C      2
 16.8   11.4  3.2     0.67857    C      2
 14.4   10.2  3.1     0.70833    C      2
 17.8   10.2  3.6     0.57303    C      2
 10.9    8.8  2.4     0.80734    B      2
 15.7   13.6  3.8     0.86624    B      2
 15.0   11.8  3.5     0.78667    B      2
  9.8    7.1  2.7     0.72449    B      2
 17.9   12.7  4.7     0.70950    A      2
 17.3   12.4  3.9     0.71676    A      2
 21.7   17.8  4.2     0.82028    A      2
   *    15.6  5.2      *         A      2
 18.1   12.1  4.5     0.66851    A      2
 16.4   11.7  4.9     0.71341    I      2
 14.9   10.8  4.5     0.72483    I      2
 23.3   17.0  6.3     0.72961    I      2
 18.6   14.6  5.7     0.78495    I      2
 21.1   15.6  6.1     0.73934    I      2
 15.0   10.2  5.9     0.68000    I      2

DATA: Tree #2

length width stem lobe ratio cluster tree TREE.NO
 11.6   8.5   3.8     0.73276   I      2
 17.4  11.1   5.7     0.63793   H      2
 14.1   9.3   4.3     0.65957   H      2
 19.7  14.4   5.4     0.73096   H      2
 16.3  11.2   3.9     0.68712   H      2
 19.6  15.4   5.0     0.78571   H      2
 13.4  10.1   3.8     0.75373   H      2
 20.7  16.7   5.3     0.80676   H      2

Box plots of lengths by cluster number (for tree #2)

Box plots of ratios by cluster number (for tree #2)

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