The ___ of recursion is the number of times a function calls itself.

Recursion is a programming technique where a function calls itself directly or indirectly in its definition. It is used to solve problems that can be broken down into smaller, similar subproblems.

A recursive function consists of two parts: the base case and the recursive case. The base case is a condition that stops the recursion, so the function does not call itself indefinitely. The recursive case is where the function calls itself, typically with smaller arguments or reduced input.

The depth of recursion is the number of times a function calls itself before reaching the base case. It is an important aspect to consider in recursive algorithms, as it can impact the efficiency, memory, and the running time of the program.

Let's consider the classic example of calculating the factorial of a number using recursion. The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. Here is a recursive function to calculate the factorial of a number: \( \text{factorial}(n) = \begin{cases} 1 & \text{if} \ n = 0 \\ n * \text{factorial}(n - 1) & \text{if} \ n > 0 \end{cases} \) In this function, the base case is when n equals 0, and the recursive case is when n is greater than 0. Let's calculate the factorial of 3 using this function to see the depth of recursion: \( \text{factorial}(3) = 3 * \text{factorial}(2) \\ \text{factorial}(2) = 2 * \text{factorial}(1) \\ \text{factorial}(1) = 1 * \text{factorial}(0) \\ \text{factorial}(0) = 1 \\ \) As we can see, the function called itself 3 times (when n was 3, 2, and 1) before reaching the base case when n was 0. Therefore, the depth of recursion for this example is 3.