In Algorithm Design Lab, time complexity of a recursive solution refers to the amount of computational time required by an algorithm as the size of its input increases. A recursive algorithm solves a problem by calling itself on smaller instances of the same problem until it reaches a base case. Although recursion can make an algorithm simple, elegant, and easier to understand, its execution time depends strongly on the number of recursive calls and the amount of work performed during each call. Therefore, analysing the time complexity of a recursive solution is essential for determining its efficiency and scalability. A recursive algorithm generally consists of two important parts: a base case and a recursive case. The base case terminates the recursion when the problem becomes sufficiently small. The recursive case reduces the original problem into one or more smaller subproblems and invokes the same algorithm on those subproblems. For example, consider a simple recursive function for calculating the factorial of a number: ```text Factorial(n): if n == 0: return 1 else: return n * Factorial(n-1) ``` For an input of size n, the function makes one recursive call with an input of size n−1. Each call performs a constant amount of additional work apart from the recursive call. The running time can therefore be expressed by the recurrence relation: T(n) = T(n−1) + O(1) Expanding the recurrence gives: T(n) = T(n−2) + O(1) + O(1) and eventually: T(n) = T(0) + nO(1) Thus, the time complexity is O(n). The recursion depth is also O(n), so the algorithm requires O(n) auxiliary stack space. A more important example is the recursive solution for binary search. Binary search operates on a sorted array and compares the target value with the middle element. If the target is not found, the algorithm recursively searches only one half of the array. Its recurrence relation is: T(n) = T(n/2) + O(1) After repeatedly dividing the problem size by two,