Factorial Using Recursion in DSA Typescript - Time & Space Complexity

Time Complexity: Factorial Using Recursion

O(n)

Understanding Time Complexity

We want to understand how the time to calculate a factorial grows as the number gets bigger.

How many steps does the computer take when using recursion to find factorial?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function factorial(n: number): number {
  if (n <= 1) return 1;
  return n * factorial(n - 1);
}
    

This code calculates the factorial of a number by calling itself with smaller numbers until it reaches 1.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive call to factorial with n-1.
• How many times: Exactly n times until reaching the base case.

How Execution Grows With Input

Each time we call factorial, it does one multiplication and calls itself with a smaller number.

Input Size (n)	Approx. Operations
10	About 10 multiplications and calls
100	About 100 multiplications and calls
1000	About 1000 multiplications and calls

Pattern observation: The number of steps grows directly with n, so doubling n doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to compute factorial grows in a straight line as the input number gets bigger.

Common Mistake

[X] Wrong: "Recursion makes factorial calculation take much longer than looping."

[OK] Correct: Both recursion and looping here do the same number of steps, so their time grows the same way.

Interview Connect

Understanding how recursion affects time helps you explain your code clearly and shows you know how to reason about performance.

Self-Check

"What if we used a loop instead of recursion? How would the time complexity change?"

Practice

1. What does the factorial of a number represent in mathematics?

easy

A. The product of all positive integers up to that number

B. The sum of all positive integers up to that number

C. The number multiplied by 2

D. The number divided by 2

Solution

1. Step 1: Understand factorial definition

   Factorial of a number n (written as n!) is the product of all positive integers from 1 to n.

2. Step 2: Compare options

   Only the product of all positive integers up to that number correctly describes factorial as a product, not sum or other operations.

3. Final Answer:

   The product of all positive integers up to that number -> Option A

4. Quick Check:

   Factorial = product of integers [OK]

Hint: Factorial means multiply all numbers down to 1 [OK]

Common Mistakes:

• Confusing factorial with sum
• Thinking factorial divides
• Mixing factorial with doubling

2. Which of the following is the correct base case for a recursive factorial function in TypeScript?

easy

A. if (n === 0) return 1;

B. if (n === 1) return 0;

C. if (n === 0) return 0;

D. if (n === 1) return n + 1;

Solution

1. Step 1: Identify factorial base case

   Factorial of 0 is defined as 1, so the base case should return 1 when n is 0.

2. Step 2: Check options

   Only if (n === 0) return 1; returns 1 when n is 0, which is correct base case for recursion.

3. Final Answer:

   if (n === 0) return 1; -> Option A

4. Quick Check:

   Base case: 0! = 1 [OK]

Hint: Base case for factorial is 0 returns 1 [OK]

Common Mistakes:

• Returning 0 for base case
• Using wrong base condition
• Returning n+1 incorrectly

3. What is the output of this TypeScript code?

function factorial(n: number): number {
  if (n === 0) return 1;
  return n * factorial(n - 1);
}
console.log(factorial(4));

medium

A. 120

B. 10

C. 16

D. 24

Solution

1. Step 1: Trace recursive calls for factorial(4)

   factorial(4) = 4 * factorial(3) factorial(3) = 3 * factorial(2) factorial(2) = 2 * factorial(1) factorial(1) = 1 * factorial(0) factorial(0) = 1 (base case)

2. Step 2: Calculate values bottom-up

   factorial(1) = 1 * 1 = 1 factorial(2) = 2 * 1 = 2 factorial(3) = 3 * 2 = 6 factorial(4) = 4 * 6 = 24

3. Final Answer:

   24 -> Option D

4. Quick Check:

   4! = 24 [OK]

Hint: Multiply down from n to 1 recursively [OK]

Common Mistakes:

• Stopping recursion too early
• Adding instead of multiplying
• Wrong base case

4. Identify the error in this recursive factorial function:

function factorial(n: number): number {
  if (n === 1) return 1;
  return n * factorial(n - 1);
}
console.log(factorial(0));

medium

A. No error, output is 1

B. Infinite recursion for input 0

C. Syntax error due to missing semicolon

D. Returns 0 for input 0

Solution

1. Step 1: Analyze base case for input 0

   Base case triggers only when n === 1, so factorial(0) calls factorial(-1), then factorial(-2), and so on infinitely.

2. Step 2: Understand recursion behavior

   Without base case for 0, recursion never stops for input 0, causing infinite recursion (stack overflow).

3. Final Answer:

   Infinite recursion for input 0 -> Option B

4. Quick Check:

   Missing base case for 0 causes infinite recursion [OK]

Hint: Always include base case for 0 in factorial [OK]

Common Mistakes:

• Missing base case for 0
• Assuming 1 covers 0
• Ignoring negative inputs

5. How can you modify the recursive factorial function to handle negative inputs gracefully in TypeScript?

hard

A. Return 0 for all negative inputs inside recursion

B. Remove the base case and let recursion handle negatives

C. Add a check to return -1 for negative inputs before recursion

D. Use a loop instead of recursion for negatives

Solution

1. Step 1: Understand factorial domain

   Factorial is not defined for negative numbers, so function should handle negatives before recursion.

2. Step 2: Add input validation

   Adding a condition to return -1 (or error) for negative inputs prevents invalid recursion calls.

3. Final Answer:

   Add a check to return -1 for negative inputs before recursion -> Option C

4. Quick Check:

   Validate input before recursion to handle negatives [OK]

Hint: Check negatives first, return error code [OK]

Common Mistakes:

• Ignoring negative inputs
• Letting recursion run on negatives
• Returning wrong values for negatives