Tree Growth Laboratory

In this lab, we will look at data for how the age of a tree is related to its diameter. the original measurements are made with some minor damage to the trees and significant time in the lab. The hope, however, is that with a base set of data, future determinations of age can be made without damaging the trees. that is, by using the base data, we can determine the age of a tree simply by measuring its diameter.

The following data was collected by Dr. Gary Walker and his students at Appalachian State University. Dr. Walker's interest lies in cliff face ecology and the base tree data is taken from Northern White Cedar in the several areas around eastern Tennessee and western North Carolina. The northern white cedar is a glacial relic, left in river gorges as the last ice age left. The data is taken by removing a core from the trees and counting the rings. Ideally, it would be nice to collect data on the age distribution without taking cores and counting rings.

The goal of this laboratory is to use the existing data to determine a relationship between the diameter and age of the trees. Based on this determined relationship, we will make predictions for the age of several trees that were not cored. These predictions will be based on two different methods and using two different functional forms for the relationship between diameter and age.

Diameter (in.)  Age  Diamter  Age  Diameter  Age  Diameter  Age 
1.0  27  2.3  38  2.4  22  3.0  56 
3.2  76  3.3  26  3.5  28  3.6  30 
3.9  18  4.0  24  4.2  44  4.4  44 
4.5  52  4.6  25  4.7  69  5.0  31 
5.2  38  5.3  56  5.4  54  5.5  25 
5.6  27  5.7  64  5.8  76  6.0  78 
6.4  67  6.7  27  7.2  76  7.5  31 
7.7  66  7.8  36  8.2  57  8.5  38 
9.0  56  10.0  60  10.3  132  10.7  40 
11.0  110  11.2  42  12.0  105  12.5  122 
13.2  56  13.8  134  14.0  106  14.1  88 
14.3  10  16.3  139  17.0  64  17.3  136 
19.0  81  20.0  96  20.6  124  22.0  139 
23.0  148  27.0  152         

Table 1. This data of northern white cedar was taken by Dr. Gary Walker and students iun 1986.  

Methods

To analyze this data, we want to determine a curve that fits the data in the best possible way. We will use three different methods. The first method will be the standard least squares fit for a straight line that a standard spreadsheet provides. The second method willl be an absolute value fit for a straight line, and the third will be a least squares fit, but with a curve chosen by each student. The last two methods will require the use of the excel spreadsheet provided. Record for functions and explain your rationale for the functional form used in the last method.

Making Predictions

We have calculated three different functions which approximate the data collected by Dr. Walker. Using these functions, we can make predictions for other data sets. For each function, we can enter in the diameter of a new tree and the result of the function is our "best fit approximation".

Questions

1. Compare the three fits. Which one seems to provide the best fit to the data?

2. Even though a more complex function might fit the data better, what reason might you use to explain why you might not want to use the "better fit".

3. Dr. Walker has taken additional measurements of 8 trees along the cliff face. The diameter measurements are listed in the Table below. Fill in the rest of the table with your predictions using the three diffferent methods.

Diameter  Abs. Linear  LS Linear  LS Other 
0.5       
1.8       
5.5       
11.6       
15.3       
23.2       
30.0       
78.3       

4. The approximations we have been using are all reasonable approaches to this problem, but all have their limitations. Suggest some boundaries for when the approximations should and should not be used for making predictions.

5. I have left out an important piece of information regarding the collection of the data. What might it be and why might this be important.

6. There are several ways to measure a tree. An alternative to measuring the diameter of the tree would be to measure the circumference. How might the results of the study differ and when might one of the other be appropriate?

7. Estimate the speed of growth of the trees based on the three approximations you found.

 

Follow-up

1. What do you feel was the purpose of this lab?

2. What did you learn in this lab?

3. What changes would you make to this lab?

4. Please rate the difficulty of this lab on a scale of 1 to 10.