## Concept:

# Tail recursion

## Headline

A form of single recursion suited for simple recursion removal

## Illustration

A tail recursive function *f* is of the following form:

*f(x)* =

*g(x)*, if*p(x)**f(r(x))*, otherwise

*g*,

*p*, and

*r*.

The schema variables *g*, *p*, and *r* serve these roles:

*p*is the condition of the base case.*g*is the computation of the base case.*r*is the computation for the argument of a recursive application.

Consider the following tail-recursive formulation of a function for testing ints to be even.

```
// Assume n>=0
public static boolean even(int n) {
if (n==0) return true;
else if (n==1) return false;
else return even(n-2);
}
```

The function is tail-recursive, because the function returns right away after the recursive call.

For comparison, the following formulation is not tail-recursive; rather primitive recursion is used.

```
// Assume n>=0
public static boolean even(int n) {
if (n==0) return true;
else return !even(n-1);
}
```

Every tail-recursive formulation can be easily translated into an iterative one, subject to tail-call elimination.

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