Recursively

Recursively is an adverb describing actions that are performed or definitions that are established by means of recursion. When something is done recursively, it refers to a smaller instance of itself or to previously defined steps as part of the process, until a base case is reached.

Etymology: The word derives from recursion, from Latin recursio, from recurrere meaning to run back.

In mathematics, recursive definitions specify a base case and a rule to obtain any term from earlier terms. Examples include the factorial function with n! = n×(n−1)! and 0! = 1, and the Fibonacci sequence with F(n) = F(n−1) + F(n−2) with F(0) = 0 and F(1) = 1. Recurrence relations describe objects defined by such rules.

In computer science, a function is recursive if it calls itself. Each invocation adds a frame on the call stack and reduces the problem toward a base case. Recursive techniques are natural for processing trees or performing divide-and-conquer algorithms, such as quicksort or depth-first search. They can be elegant but may incur overhead; in some languages, tail recursion is optimized to avoid growing the stack.

In other domains, recursion appears in linguistics as recursive embedding of clauses, and in computer science education as a teaching example for recursion concepts. The term recursively is often used to describe algorithms, definitions, or procedures that operate by applying the same rule to progressively simpler instances.