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The curiously recurring template pattern (CRTP) is an idiom, originally in C++, in which a class X derives from a class template instantiation using X itself as a template argument. [1] More generally it is known as F-bound polymorphism , and it is a form of F -bounded quantification .
One example is tree traversal as in depth-first search; though both recursive and iterative methods are used, [17] they contrast with list traversal and linear search in a list, which is a singly recursive and thus naturally iterative method. Other examples include divide-and-conquer algorithms such as Quicksort, and functions such as the ...
Another example is a similar singly linked type in Java: class List < E > { E value ; List < E > next ; } This indicates that non-empty list of type E contains a data member of type E, and a reference to another List object for the rest of the list (or a null reference to indicate that this is the end of the list).
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 ...
A recursive step — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ancestor. One's ancestor is either: One's parent (base case), or; One's parent's ancestor (recursive step). The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as ...
Recursion allows direct implementation of functionality defined by mathematical induction and recursive divide and conquer algorithms. Here is an example of a recursive function in C/C++ to find Fibonacci numbers:
This enumeration uses the definitions of the primitive recursive functions (which are essentially just expressions with the composition and primitive recursion operations as operators and the basic primitive recursive functions as atoms), and can be assumed to contain every definition once, even though a same function will occur many times on ...
In C++, classes can be forward-declared if you only need to use the pointer-to-that-class type (since all object pointers are the same size, and this is what the compiler cares about). This is especially useful inside class definitions, e.g. if a class contains a member that is a pointer (or a reference) to another class.