Mutual recursion

In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational objects, such as functions or datatypes, 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 datatypes are naturally mutually recursive. The most important basic example of a datatype 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: f: [t[1], ..., t[k]] t: v f 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: t: v [t[1], ..., t[k]] 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 datatypes can be mutually recursively defined as follows, allowing empty trees: datatype 'a tree = Empty | Node of 'a * 'a forest and 'a forest = Nil | Cons of 'a tree * 'a forest Just as algorithms on recursive datatypes 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.