A recursive algorithm is a fundamental programming concept where a function calls itself to solve problems. It works by breaking down a complex problem into smaller, similar subproblems. Every recursive algorithm must have a base case to stop the recursion and a recursive step that reduces the problem size.
The factorial function is a perfect example of recursion. To calculate factorial of n, we multiply n by the factorial of n minus 1. The base case is when n equals 0, which returns 1. Let's trace through factorial of 4: it calls factorial of 3, which calls factorial of 2, and so on until we reach the base case.
The Fibonacci sequence is another classic recursive example. Each Fibonacci number is the sum of the two preceding numbers. The base cases are F of 0 equals 0 and F of 1 equals 1. This creates a binary tree of recursive calls, where each function call branches into two more calls, demonstrating how recursion can lead to exponential complexity.
Binary tree traversal demonstrates recursion's power in handling hierarchical data structures. In inorder traversal, we recursively visit the left subtree, then the root, then the right subtree. The recursive nature perfectly matches the tree's structure, where each subtree is itself a smaller tree that can be processed the same way.
To master recursion, remember these key principles: always define a clear base case to stop the recursion, ensure each recursive call makes progress toward the base case, and keep recursive calls simple and focused. Consider the efficiency of your recursive solution, as some like naive Fibonacci can be exponentially slow. Use recursion when the problem naturally has a recursive structure, like trees or graphs, but prefer iteration for simple counting or looping tasks.