24. Recursion

A child couldn't sleep, so her mother told her a story about a little frog,
  who couldn't sleep, so the frog's mother told her a story about a little bear,
    who couldn't sleep, so the bear's mother told her a story about a little weasel...
      who fell asleep.
    ...and the little bear fell asleep;
  ...and the little frog fell asleep;
...and the child fell asleep.

(Source: http://everything2.com/title/recursion)

Recursion is an object or process that is defined in terms of itself. Mathematical patterns such as factorials and the Fibonacci series are recursive. Documents that can contain other documents, which themselves can contain other documents, are recursive. Fractal images, and even certain biological processes are recursive in how they work.

24.1. Where is Recursion Used?

Documents, such as web pages, are naturally recursive. For example, Figure 20.1 shows a simple web document.

../../_images/webpage1.png

Figure 20.1: Web page

That web document can be contained in a “box,” which can help layout the page as shown in Figure 20.2.

../../_images/webpage2.png

Figure 20.2: Web page with tables

This works recursively. Each box can contain a web page, that can have a box, which could contain another web page as shown in Figure 20.3.

../../_images/webpage3.png

Figure 20.3: Web page with recursion

Recursive functions are often used with advanced searching and sorting algorithms. We’ll show some of that here and if you take a “data structures” class you will see a lot more of it.

Even if a person does not become a programmer, understanding the concept of recursive systems is important. If there is a business need for recursive table structures, documents, or something else, it is important to know how to specify this to the programmer up front.

For example, a person might specify that a web program for recipes needs the ability to support ingredients and directions. A person familiar with recursion might state that each ingredient could itself be a recipes with other ingredients (that could be recipes.) The second system is considerably more powerful.

24.2. How is Recursion Coded?

In prior chapters, we have used functions that call other functions. For example:

Functions calling other functions
1
2
3
4
5
6
7
8
def f():
    g()
    print("f")

def g():
    print("g")

f()

It is also possible for a function to call itself. A function that calls itself is using a concept called recursion. For example:

Recursion
1
2
3
4
5
def f():
    print("Hello")
    f()

f()

The example above will print Hello and then call the f() function again. Which will cause another Hello to be printed out and another call to the f() function. This will continue until the computer runs out of something called stack space. When this happens, Python will output a long error that ends with:

RuntimeError: maximum recursion depth exceeded

The computer is telling you, the programmer, that you have gone too far down the rabbit hole.

24.3. Controlling Recursion Depth

To successfully use recursion, there needs to be a way to prevent the function from endlessly calling itself over and over again. The example below counts how many times it has been called, and uses an if statement to exit once the function has called itself ten times.

Controlling recursion levels
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
def f(level):
    # Print the level we are at
    print("Recursion call, level",level)
    # If we haven't reached level ten...
    if level < 10:
        # Call this function again
        # and add one to the level
        f(level+1)

# Start the recursive calls at level 1
f(1)
Output
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
Recursion call, level 1
Recursion call, level 2
Recursion call, level 3
Recursion call, level 4
Recursion call, level 5
Recursion call, level 6
Recursion call, level 7
Recursion call, level 8
Recursion call, level 9
Recursion call, level 10

24.4. Recursion In Mathematics

24.4.1. Recursion Factorial Calculation

Any code that can be done recursively can be done without using recursion. Some programmers feel that the recursive code is easier to understand.

Calculating the factorial of a number is a classic example of using recursion. Factorials are useful in probability and statistics. For example:

Recursively, this can be described as:

Below are two example functions that calculate . The first one is non-recursive, the second is recursive.

Non-recursive factorial
1
2
3
4
5
6
7
# This program calculates a factorial
# WITHOUT using recursion
def factorial_nonrecursive(n):
    answer = 1
    for i in range(2, n + 1):
        answer = answer * i
    return answer
Recursive factorial
1
2
3
4
5
6
7
# This program calculates a factorial
# WITH recursion
def factorial_recursive(n):
    if n == 1:
        return 1
    elif n > 1:
        return n * factorial_recursive(n - 1)

The functions do nothing by themselves. Below is an example where we put it all together. This example also adds some print statements inside the function so we can see what is happening.

Trying out recursive functions
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
# This program calculates a factorial
# WITHOUT using recursion

def factorial_nonrecursive(n):
    answer = 1
    for i in range(2, n + 1):
        print(i, "*", answer, "=", i * answer)
        answer = answer * i
    return answer

print("I can calculate a factorial!")
user_input = input("Enter a number:")
n = int(user_input)
answer = factorial_nonrecursive(n)
print(answer)

# This program calculates a factorial
# WITH recursion

def factorial_recursive(n):
    if n == 1:
        return 1
    else:
        x = factorial_recursive(n - 1)
        print( n, "*", x, "=", n * x )
        return n * x

print("I can calculate a factorial!")
user_input = input("Enter a number:")
n = int(user_input)
answer = factorial_recursive(n)
print(answer)
Output
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
I can calculate a factorial!
Enter a number:7
2 * 1 = 2
3 * 2 = 6
4 * 6 = 24
5 * 24 = 120
6 * 120 = 720
7 * 720 = 5040
5040
I can calculate a factorial!
Enter a number:7
2 * 1 = 2
3 * 2 = 6
4 * 6 = 24
5 * 24 = 120
6 * 120 = 720
7 * 720 = 5040
5040

24.4.2. Recursive Expressions

Say you have a mathematical expression like this:

\(f_{n} = \begin{cases} 6 & \text{if } n = 1, \\ \frac{1}{2}f_{n-1}+4 & \text{if } n > 1. \end{cases}\)

Looks complicated, but it just means that if \(n=1\) we are working with \(f_{1}\). That function returns a 6.

For \(f_{2}\) we return \(\frac{1}{2}f_{1}+4\).

The code would start with:

1
def f(n):

Then we need to add that first case:

1
2
3
def f(n):
    if n == 1:
        return 6

See how closely if follows the mathematical notation? Now for the rest:

1
2
3
4
5
def f(n):
    if n == 1:
        return 6
    elif n > 1:
        return (1 / 2) * f(n - 1) + 4

Converting these types of mathematical expressions to code is straight forward. But we’d better try it out in a full example:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
def f(n):
    if n == 1:
        return 6
    elif n > 1:
        return (1 / 2) * f(n - 1) + 4


def main():
    result = f(10)
    print(result)


main()

24.5. Recursive Graphics

24.5.1. Recursive Rectangles

Recursion is great to work with structured documents that are themselves recursive. For example, a web document can have a table divided into rows and columns to help with layout. One row might be the header, another row the main body, and finally the footer. Inside a table cell, might be another table. And inside of that can exist yet another table.

Another example is e-mail. It is possible to attach another person’s e-mail to a your own e-mail. But that e-mail could have another e-mail attached to it, and so on.

Can we visually see recursion in action in one of our Pygame programs? Yes! Figure 19.4 shows an example program that draws a rectangle, and recursively keeps drawing rectangles inside of it. Each rectangle is 20% smaller than the parent rectangle. Look at the code. Pay close attention to the recursive call in the recursive_draw function.

../../_images/rectangles.png

Figure 20.4: Recursive Rectangles

recursive_rectangles.py
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
"""
Recursive Rectangles
"""
import arcade

SCREEN_WIDTH = 800
SCREEN_HEIGHT = 500


def draw_rectangle(x, y, width, height):
    """ Recursively draw a rectangle, each one a percentage smaller """

    # Draw it
    arcade.draw_rectangle_outline(x, y, width, height, arcade.color.BLACK)

    # As long as we have a width bigger than 1, recursively call this function with a smaller rectangle
    if width > 1:
        # Draw the rectangle 90% of our current size
        draw_rectangle(x, y, width * .9, height * .9)


class MyWindow(arcade.Window):
    """ Main application class. """

    def __init__(self, width, height):
        super().__init__(width, height)

        arcade.set_background_color(arcade.color.WHITE)

    def on_draw(self):
        """ Render the screen. """
        arcade.start_render()

        # Find the center of our screen
        center_x = SCREEN_WIDTH / 2
        center_y = SCREEN_HEIGHT / 2

        # Start our recursive calls
        draw_rectangle(center_x, center_y, SCREEN_WIDTH, SCREEN_HEIGHT)


def main():

    MyWindow(SCREEN_WIDTH, SCREEN_HEIGHT)
    arcade.run()


if __name__ == "__main__":
    main()

24.5.2. Fractals

Fractals are defined recursively. Here is a very simple fractal, showing how it changes depending on how “deep” the recursion goes.

../../_images/recursive_h_00.png

Figure 20.5: Recursive Fractal Level 0

../../_images/recursive_h_01.png

Figure 20.6: Recursive Fractal Level 1

../../_images/recursive_h_02.png

Figure 20.7: Recursive Fractal Level 2

../../_images/recursive_h_05.png

Figure 20.8: Recursive Fractal Level 5

Here is the source code for the “H” fractal:

recursive_h.py
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
"""
Recursive H's
"""
import arcade

SCREEN_WIDTH = 800
SCREEN_HEIGHT = 500

RECURSION_DEPTH = 0


def draw_h(x, y, width, height, count):
    """ Recursively draw an H, each one a half as big """

    # Draw the H
    # Draw cross-bar
    arcade.draw_line(x + width * .25, height / 2 + y,
                     x + width * .75, height / 2 + y, arcade.color.BLACK)
    # Draw left side
    arcade.draw_line(x + width * .25, height * .5 / 2 + y,
                     x + width * .25, height * 1.5 / 2 + y, arcade.color.BLACK)
    # Draw right side
    arcade.draw_line(x + width * .75, height * .5 / 2 + y,
                     x + width * .75, height * 1.5 / 2 + y, arcade.color.BLACK)

    # As long as we have a width bigger than 1, recursively call this function with a smaller rectangle
    if count > 0:
        count -= 1
        # Draw the rectangle 90% of our current size
        # Draw lower left
        draw_h(x, y, width / 2, height / 2, count)
        # Draw lower right
        draw_h(x + width / 2, y, width / 2, height / 2, count)
        # Draw upper left
        draw_h(x, y + height / 2, width / 2, height / 2, count)
        # Draw upper right
        draw_h(x + width / 2, y + height / 2, width / 2, height / 2, count)


class MyWindow(arcade.Window):
    """ Main application class. """

    def __init__(self, width, height):
        super().__init__(width, height)

        arcade.set_background_color(arcade.color.WHITE)

    def on_draw(self):
        """ Render the screen. """
        arcade.start_render()

        # Start our recursive calls
        draw_h(0, 0, SCREEN_WIDTH, SCREEN_HEIGHT, RECURSION_DEPTH)


def main():
    MyWindow(SCREEN_WIDTH, SCREEN_HEIGHT)
    arcade.run()


if __name__ == "__main__":
    main()

You can explore fractals on-line:

If you want to program your own fractals, you can get ideas of easy fractals by looking at Chapter 8 of The Nature of Code by Daniel Shiffman.

24.6. Recursive Mazes

There are maze generation algrithms. Wikipedia has a nice Maze generation algorithm article that details some. One way is the recursive division method.

The algorithm is described below. Images are from Wikipedia.

../../_images/chamber.svg

Begin with the maze’s space with just the outside walls. Call this a chamber.

../../_images/chamber-division.svg

Divide the chamber with one random horizontal wall, and one random vertical wall.

../../_images/chamber-divided.svg

Pick three of the four walls and put a randomly position opening in it.

../../_images/chamber-subdivision.svg

Subdivide each of the four sections. Call each a chamber. Recursively call this function. In this image, the top left chamber has been subdivided.

../../_images/chamber-finished.svg

Finished maze.

This method results in mazes with long straight walls crossing their space, making it easier to see which areas to avoid.

../../_images/recursive_maze.gif

Recursive maze generation. Source: Wikipedia Maze generation algorithm.

Here is sample Python code that creates a maze using this method:

Recursive Maze Example
  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
import random

# These constants are used to determine what should be stored in the grid if we have an empty
# space or a filled space.
EMPTY = "   "
WALL = "XXX"

# Maze must have an ODD number of rows and columns.
# Walls go on EVEN rows/columns.
# Openings go on ODD rows/columns
MAZE_HEIGHT = 51
MAZE_WIDTH = 51


def create_grid(width, height):
    """ Create an empty grid. """
    grid = []
    for row in range(height):
        grid.append([])
        for column in range(width):
            grid[row].append(EMPTY)
    return grid


def print_maze(maze):
    """ Print the maze. """

    # Loop each row, but do it in reverse so 0 is at the bottom like we expect
    for row in range(len(maze) - 1, -1, -1):
        # Print the row/y number
        print(f"{row:3} - ", end="")

        # Loop the row and print the content
        for column in range(len(maze[row])):
            print(f"{maze[row][column]}", end="")

        # Go down a line
        print()

    # Print the column/x at the bottom
    print("     ", end="")
    for column in range(len(maze[0])):
        print(f"{column:3}", end="")
    print()


def create_outside_walls(maze):
    """ Create outside border walls."""

    # Create left and right walls
    for row in range(len(maze)):
        maze[row][0] = WALL
        maze[row][len(maze[row])-1] = WALL

    # Create top and bottom walls
    for column in range(1, len(maze[0]) - 1):
        maze[0][column] = WALL
        maze[len(maze[0]) - 1][column] = WALL


def create_maze(maze, top, bottom, left, right):
    """
    Recursive function to divide up the maze in four sections
    and create three gaps.
    Walls can only go on even numbered rows/columns.
    Gaps can only go on odd numbered rows/columns.
    Maze must have an ODD number of rows and columns.
    """

    # Figure out where to divide horizontally
    start_range = bottom + 2
    end_range = top - 1
    y = random.randrange(start_range, end_range, 2)

    # Do the division
    for column in range(left + 1, right):
        maze[y][column] = WALL

    # Figure out where to divide vertically
    start_range = left + 2
    end_range = right - 1
    x = random.randrange(start_range, end_range, 2)

    # Do the division
    for row in range(bottom + 1, top):
        maze[row][x] = WALL

    # Now we'll make a gap on 3 of the 4 walls.
    # Figure out which wall does NOT get a gap.
    wall = random.randrange(4)
    if wall != 0:
        gap = random.randrange(left + 1, x, 2)
        maze[y][gap] = EMPTY

    if wall != 1:
        gap = random.randrange(x + 1, right, 2)
        maze[y][gap] = EMPTY

    if wall != 2:
        gap = random.randrange(bottom + 1, y, 2)
        maze[gap][x] = EMPTY

    if wall != 3:
        gap = random.randrange(y + 1, top, 2)
        maze[gap][x] = EMPTY

    # Print what's going on
    print(f"Top/Bottom: {top}, {bottom} Left/Right: {left}, {right} Divide: {x}, {y}")
    print_maze(maze)
    print()

    # If there's enough space, to a recursive call.
    if top > y + 3 and x > left + 3:
        create_maze(maze, top, y, left, x)

    if top > y + 3 and x + 3 < right:
        create_maze(maze, top, y, x, right)

    if bottom + 3 < y and x + 3 < right:
        create_maze(maze, y, bottom, x, right)

    if bottom + 3 < y and x > left + 3:
        create_maze(maze, y, bottom, left, x)


def main():

    # Create the blank grid
    maze = create_grid(MAZE_WIDTH, MAZE_HEIGHT)

    # Fill in the outside walls
    create_outside_walls(maze)

    # Start the recursive process
    create_maze(maze, MAZE_HEIGHT - 1, 0, 0, MAZE_WIDTH - 1)


if __name__ == "__main__":
    main()