# Tail recursion

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
for suitable 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.
We use Language:Java for illustration.

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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