1. Write a recursive function called intpow that given a number, x, and an integer, n, will compute x ^ n. You must write this function recursively to get full credit. Be sure to put it in a program with several test cases to test that your function works correctly.

2. Write a recursive function to compute the factorial of an integer. The factorial of 0 is 1. The factorial of any integer, n, greater than zero is n times the factorial of n−1. Write a program that tests your factorial function by asking the user to enter an integer and printing the factorial of that integer. Be sure your program has a main function. Comment your code with the base case and recursive case in your recursive function.

3. Write a recursive function that computes the length of a string. You cannot use the len function while computing the length of the string. You must rely on the function you are writing. Put this function in a program that prompts the user to enter a string and then prints the length of that string.

Fig. 3.9 A Tree

4. Write a recursive function that takes a string like “abcdefgh” and returns “badcfehg”. Call this function swap since it swaps every two elements of the original string. Put this function in a program and call it with at least a few test cases.

5. Write a recursive function that draws a tree. Call your function drawBranch. Pass it a turtle to draw with, an angle, and the number of littler branches to draw like the tree that appears in Fig. 3.9. Each time you recursively call this function you can decrease the number of branches and the angle. Each littler branch is drawn at some angle from the current branch so your function can change the angle of the turtle by turning left or right. When your number of branches gets to zero, you can draw a leaf as a little green square. If you make the width of the turtle line thicker for bigger branches and smaller for littler branches, you'll get a nice tree. You might write one more function called drawTree that will set up everything (except the turtle) to draw a nice tree. Put this function in a program that draws at least one tree. HINT: In your drawBranch function, after you have drawn the branch (and all sub-branches) you will want to return the turtle to the original position and direction you started at. This is necessary so after calling drawBranch you will know where the turtle is located. If you don't return it to its original position, the turtle will end up stranded out at a leaf somewhere.

6. Write a recursive function that draws a circular spiral. To do this, you'll need to use polar coordinates. Polar coordinates are a way of specifying any point in the plane with an angle and a radius. Zero degrees goes to the right and the angles go counter-clockwise in a circle. With an angle and a radius, any point in the plane can be described. To convert an angle, a, and radius, r, from polar coordinates to Cartesian coordinates you would use sine and cosine. You must import the math module. Then x = r * math.cos (a) and y = r * math.sin (a).

The drawSpiral function will be given a radius for the sprial. To get a circular spiral, every recursive call to the drawSpiral function must decrease the radius just a bit and increase the angle. You convert the angle and the radius to its (x,y) coordinate equivalent and then draw a line to that location. You must also pass an (x,y) coordinate to the drawSpiral function for the center point of your spiral. Then, any coordinates you compute will be added to the center (x,y). You can follow the square spiral example in the text. Put this code in a program that draws a spiral to the screen.

7. Write a program to gather performance data for the reverse function found in this chapter. Write an XML file in the plot format found in this text to visualize that performance data. Because this function is recursive, keep your data size small and just gather data for string sizes of 1–10. This will help you visualize your result. What is the complexity of this reverse function? Put a comment at the top of your program stating the complexity of reverse in big-Oh notation. Justify your answer by analyzing the code found in the reverse function.

8. Rewrite the program in Sect. 3.7.4 to use an index that approaches the length of the list instead of an index that approaches zero. Then write a main function that thoroughly tests your new reverse function on lists. You must test it on both simple and more complex examples of lists to test it thoroughly.

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