Mutual recursion

In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational objects, such as functions or data types, are defined in terms of each other. Mutual recursion is very common in functional programming and in some problem domains, such as recursive descent parsers, where the data types are naturally mutually recursive, but is uncommon in other domains.

The most important basic example of a data type that can be defined by mutual recursion is a tree, which can be defined mutually recursively in terms of a forest (a list of trees). Symbolically:

A forest f consists of a list of trees, while a tree t consists of a pair of a value v and a forest f (its children). This definition is elegant and easy to work with abstractly (such as when proving theorems about properties of trees), as it expresses a tree in simple terms: a list of one type, and a pair of two types. Further, it matches many algorithms on trees, which consist of doing one thing with the value, and another thing with the children.

This mutually recursive definition can be converted to a singly recursive definition by inlining the definition of a forest:

A tree t consists of a pair of a value v and a list of trees (its children). This definition is more compact, but somewhat messier: a tree consists of a pair of one type and a list of another, which require disentangling to prove results about.

In Standard ML, the tree and forest data types can be mutually recursively defined as follows, allowing empty trees:

Just as algorithms on recursive data types can naturally be given by recursive functions, algorithms on mutually recursive data structures can be naturally given by mutually recursive functions. Common examples include algorithms on trees, and recursive descent parsers. As with direct recursion, tail call optimization is necessary if the recursion depth is large or unbounded, such as using mutual recursion for multitasking. Note that tail call optimization in general (when the function called is not the same as the original function, as in tail-recursive calls) may be more difficult to implement than the special case of tail-recursive call optimization, and thus efficient implementation of mutual tail recursion may be absent from languages that only optimize tail-recursive calls. In languages such as Pascal that require declaration before use, mutually recursive functions require forward declaration, as a forward reference cannot be avoided when defining them.

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