Recursion

[aayush_005]

The Basic Idea

We have already seen how, in a well designed C++ program, many function definitions include calls to other functions (for example, in the last lecture the definition of "assign_list(...)" included a call to "assign_new_node(...)"). A function is recursive (or has a recursive definition) if the definition includes a call to itself.

Recursion is a familiar idea in mathematics and logic. For example, the natural numbers themselves are usually defined recursively. Very roughly speaking, the definition is:
0 is a natural number.

if n is a natural number then s(n) (i.e. n+1) is a natural number, where s is the "successor function".


In this context, the notion of recursion is clearly related to the notion of mathematical induction. Notice also that the above definition includes a non-recursive part or base case (the statement that 0 is a natural number).

Another familiar mathematical example of a recursive function is the factorial function "!". Its definition is:
0! = 1

for all n > 0, n! = nx(n-1)!


Thus, by repeatedly using the definition, we can work out that
6! = 6x5! = 6x5x4! = 6x5x4x3! = 6x5x4x3x2! = 6x5x4x3x2x1! = 6x5x4x3x2x1x1 = 720

Again, notice that the definition of "!" includes both a base case (the definition of 0!) and a recursive part.
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8.2 A Simple Example

The following program includes a call to the recursively defined function "print_backwards()", which inputs a series of characters from the keyboard, terminated with a full-stop character, and then prints them backwards on the screen.
#include<iostream>
using namespace std;

void print_backwards();

int main()
{
print_backwards();
cout << "\n";

return 0;
}

void print_backwards()
{
char character;

cout << "Enter a character ('.' to end program): ";
cin >> character;
if (character != '.')
{
print_backwards();
cout << character;
}
}
Program 8.2.1

A typical input/output session is:
Enter a character ('.' to end program): H
Enter a character ('.' to end program): i
Enter a character ('.' to end program): .
iH

We will examine how this function works in more detail in the next section. But notice that the recursive call to "print_backwards()" (within its own definition) is embedded in an "if" statement. In general, recursive definitions must always use some sort of branch statement with at least one non-recursive branch, which acts as the base case of the definition. Otherwise they will "loop forever". In Program 8.2.1 the base case is in the implicit "else" part of the "if" statement. We could have written the function as follows:
void print_backwards()
{
char character;

cout << "Enter a character ('.' to end program): ";
cin >> character;
if (character != '.')
{
print_backwards();
cout << character;
}
else
{
;
}
}

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8.3 The Mechanics of a Recursive Call

It is easy to see why Program 8.2.1 works with the aid of a few diagrams. When the main program executes, it begins with a call to "print_backwards()". At this point space is set aside in the computer's memory to execute this call (and in other cases in which to make copies of the value parameters). This space is represented as a box in Figure 8.3.1a:


Figure 8.3.1a

The internal execution of this call begins with a character input, and then a second call to "print_backwards()" (at this point, nothing has been output to the screen). Again, space is set aside for this second call:


Figure 8.3.1b

The process repeats, but inside the third call to "print_backwards()" a full-stop character is input, thus allowing the third call to terminate with no further function calls:


Figure 8.3.1c

This allows the second call to "print_backwards()" to terminate by outputting an "i" character, which in turn allows the first call to terminate by outputting an "H" character:


Figure 8.3.1d

Technically speaking, C++ arranges the memory spaces needed for each function call in a stack. The memory area for each new call is placed on the top of the stack, and then taken off again when the execution of the call is completed. In the example above, the stack goes through the following sequence:


Figure 8.3.2

C++ uses this stacking principle for all nested function calls - not just for recursively defined functions. A stack is an example of a "last in/first out" structure (as opposed to, for example, a queue, which is a "first in/first out" structure).

[stevebucknor1]

Recursion is a familiar idea in mathematics and logic. For example, the natural numbers themselves are usually defined recursively. Very roughly speaking, the definition is: 0 is a natural number. if n is a natural number then s(n) (i.e. n+1) is a natural number, where s is the "successor function". This is the basic fundamental of this.

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