Recursion Concept and Call Stack Visualization in DSA Javascript - Time & Space Complexity

Time Complexity: Recursion Concept and Call Stack Visualization

O(n)

Understanding Time Complexity

When using recursion, we want to know how many times the function calls itself as the input grows.

We ask: How does the work increase when the input gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following recursive function.

function factorial(n) {
  if (n === 0) return 1;
  return n * factorial(n - 1);
}
    

This function calculates the factorial of a number by calling itself with smaller values until it reaches zero.

Identify Repeating Operations

Look for repeated calls or loops.

• Primary operation: Recursive call to factorial with n - 1
• How many times: Exactly n times until reaching 0

How Execution Grows With Input

Each time we call factorial, it calls itself once with a smaller number until zero.

Input Size (n)	Approx. Operations
10	11 calls
100	101 calls
1000	1001 calls

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

Final Time Complexity

Time Complexity: O(n)

This means the function does work proportional to the input size, calling itself once per number down to zero.

Common Mistake

[X] Wrong: "Recursion always makes the program slow because it repeats work many times."

[OK] Correct: In this case, each call is unique and happens once, so the work grows linearly, not exponentially.

Interview Connect

Understanding how recursion grows helps you explain your code clearly and shows you can reason about performance, a key skill in interviews.

Self-Check

"What if the function called itself twice each time (like in Fibonacci)? How would the time complexity change?"

Practice

1. What happens when a recursive function calls itself without a base case?

easy

A. It returns immediately without doing anything

B. It causes an infinite loop leading to a stack overflow error

C. It executes only once and stops

D. It throws a syntax error

Solution

1. Step 1: Understand the role of base case in recursion

   The base case stops the recursion by providing a condition to end calls.

2. Step 2: Analyze what happens without a base case

   Without a base case, the function keeps calling itself endlessly, filling the call stack.

3. Final Answer:

   It causes an infinite loop leading to a stack overflow error -> Option B

4. Quick Check:

   No base case = infinite recursion = stack overflow [OK]

Hint: Always check for a base case to stop recursion [OK]

Common Mistakes:

• Thinking recursion stops automatically
• Confusing base case with recursive case
• Assuming syntax error occurs without base case

2. Which of the following is the correct syntax to define a recursive function countDown in JavaScript?

easy

A. function countDown(n) { if(n <= 0) return; countDown(n - 1); }

B. function countDown(n) { while(n > 0) countDown(n - 1); }

C. function countDown(n) { for(let i=0; i

D. function countDown(n) { if(n < 0) countDown(n + 1); }

Solution

1. Step 1: Identify correct base case and recursive call

   function countDown(n) { if(n <= 0) return; countDown(n - 1); } has a base case (if n <= 0 return) and calls itself with n-1.

2. Step 2: Check other options for syntax or logic errors

   function countDown(n) { while(n > 0) countDown(n - 1); } uses while loop causing infinite recursion; C uses for loop incorrectly; D lacks proper base case.

3. Final Answer:

   function countDown(n) { if(n <= 0) return; countDown(n - 1); } -> Option A

4. Quick Check:

   Base case + recursive call = correct syntax [OK]

Hint: Look for base case and self-call in function [OK]

Common Mistakes:

• Using loops instead of recursion
• Missing base case
• Wrong recursive call argument

3. What is the output of the following code?

function factorial(n) {
  if (n === 0) return 1;
  return n * factorial(n - 1);
}
console.log(factorial(3));

medium

A. 9

B. 1

C. 3

D. 6

Solution

1. Step 1: Trace factorial calls for n=3

   factorial(3) = 3 * factorial(2); factorial(2) = 2 * factorial(1); factorial(1) = 1 * factorial(0); factorial(0) = 1.

2. Step 2: Calculate return values step-by-step

   factorial(1) = 1 * 1 = 1; factorial(2) = 2 * 1 = 2; factorial(3) = 3 * 2 = 6.

3. Final Answer:

   6 -> Option D

4. Quick Check:

   3! = 6 [OK]

Hint: Factorial n! = n * (n-1)! [OK]

Common Mistakes:

• Confusing multiplication order
• Forgetting base case returns 1
• Calculating factorial(0) incorrectly

4. Find the error in this recursive function that calculates the sum of numbers from 1 to n:

function sum(n) {
  if (n === 1) return 1;
  return n + sum(n);
}

medium

A. The base case is missing

B. The function should return n - sum(n - 1)

C. The recursive call uses the same argument causing infinite recursion

D. The function should use a loop instead

Solution

1. Step 1: Check the recursive call argument

   The function calls sum(n) again without reducing n, causing infinite recursion.

2. Step 2: Identify correct recursive call

   It should call sum(n - 1) to reduce n and reach base case.

3. Final Answer:

   The recursive call uses the same argument causing infinite recursion -> Option C

4. Quick Check:

   Recursive call argument must change to reach base case [OK]

Hint: Recursive call argument must move toward base case [OK]

Common Mistakes:

• Calling function with same argument
• Missing base case
• Using wrong operator in return

5. Consider this recursive function that reverses a string:

function reverseStr(str) {
  if (str === "") return "";
  return reverseStr(str.substring(1)) + str[0];
}
console.log(reverseStr("code"));

What is the output and why?

hard

A. "edoc" because the function moves first character to the end recursively

B. "code" because substring does not change the string

C. "" because base case returns empty string immediately

D. SyntaxError because substring is used incorrectly

Solution

1. Step 1: Understand recursion on string substring

   The function calls itself with the string minus the first character each time.

2. Step 2: Trace the concatenation on return

   Each call adds the first character of the current string to the end of the reversed substring, reversing the string.

3. Final Answer:

   "edoc" because the function moves first character to the end recursively -> Option A

4. Quick Check:

   Recursive substring + concat reverses string [OK]

Hint: Recursion with substring(1) + first char reverses string [OK]

Common Mistakes:

• Thinking substring does not change string
• Confusing base case effect
• Assuming syntax error with substring