Recursion Base Case and Recursive Case in DSA Javascript - Time & Space Complexity

Time Complexity: Recursion Base Case and Recursive Case

O(n)

Understanding Time Complexity

When using recursion, it is important to understand how the time taken grows as the input changes.

We want to know how many times the function calls itself before stopping.

Scenario Under Consideration

Analyze the time complexity of the following recursive function.

function factorial(n) {
  if (n === 0) {
    return 1; // base case
  }
  return n * factorial(n - 1); // recursive case
}
    

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

Identify Repeating Operations

Look for repeated actions in the code.

• Primary operation: The function calls itself once each time with n reduced by 1.
• How many times: It repeats this call until n reaches 0, so about n times.

How Execution Grows With Input

Each time the function calls itself, it does a small amount of work and then calls again with a smaller number.

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

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

Final Time Complexity

Time Complexity: O(n)

This means the time taken grows in a straight line with the input size.

Common Mistake

[X] Wrong: "Recursion always means exponential time because it calls itself multiple times."

[OK] Correct: This example calls itself only once per step, so it grows linearly, not exponentially.

Interview Connect

Understanding how base and recursive cases affect time helps you explain your code clearly and shows you know how recursion works efficiently.

Self-Check

"What if the function called itself twice each time instead of once? How would the time complexity change?"

Practice

1. What is the purpose of the base case in a recursive function?

easy

A. To call the function again with a smaller input

B. To increase the input size for the next call

C. To stop the recursion and prevent infinite calls

D. To print the result of the recursion

Solution

1. Step 1: Understand what recursion does

   Recursion calls the same function repeatedly with smaller inputs.

2. Step 2: Identify the role of the base case

   The base case stops the recursion by providing a condition where no further calls happen.

3. Final Answer:

   To stop the recursion and prevent infinite calls -> Option C

4. Quick Check:

   Base case = stop recursion [OK]

Hint: Base case stops recursion to avoid infinite loops [OK]

Common Mistakes:

• Confusing base case with recursive case
• Thinking base case calls recursion again
• Ignoring base case leads to infinite recursion

2. Which of the following is the correct syntax for a base case in this recursive function to calculate factorial?

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

easy

A. n > 0

B. n === 0

C. n === 1

D. n < 0

Solution

1. Step 1: Recall factorial base case

   Factorial of 0 is 1, so base case is when n equals 0.

2. Step 2: Match condition to code

   The condition n === 0 correctly stops recursion and returns 1.

3. Final Answer:

   n === 0 -> Option B

4. Quick Check:

   Base case for factorial = n === 0 [OK]

Hint: Factorial base case is n equals zero [OK]

Common Mistakes:

• Using n > 0 as base case causes infinite recursion
• Using n < 0 which is invalid for factorial
• Using n === 1 misses factorial(0) base case

3. What will be the output of this recursive function call?

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

medium

A. 6

B. 3

C. 0

D. Error

Solution

1. Step 1: Trace recursive calls

   sumTo(3) calls 3 + sumTo(2), sumTo(2) calls 2 + sumTo(1), sumTo(1) calls 1 + sumTo(0), sumTo(0) returns 0.

2. Step 2: Calculate total sum

   sumTo(3) = 3 + 2 + 1 + 0 = 6.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   3+2+1+0 = 6 [OK]

Hint: Sum from n down to 0 adds all numbers [OK]

Common Mistakes:

• Forgetting to add current n before recursive call
• Confusing base case return value
• Assuming output is just n instead of sum

4. Find the error in this recursive function to compute the nth Fibonacci number:

function fib(n) {
  if (n === 0) return 0;
  if (n === 1) return 1;
  return fib(n - 1) + fib(n - 2);
}
console.log(fib(-1));

medium

A. Missing base case for negative n causing infinite recursion

B. Incorrect base case values for fib(0) and fib(1)

C. Wrong recursive calls should be fib(n + 1) and fib(n + 2)

D. No error, output is correct

Solution

1. Step 1: Analyze base cases and input

   Base cases cover n=0 and n=1, but input is n=-1 which is not handled.

2. Step 2: Understand recursion with negative input

   Calling fib(-1) leads to fib(-2) and fib(-3) calls infinitely, no stop condition.

3. Final Answer:

   Missing base case for negative n causing infinite recursion -> Option A

4. Quick Check:

   Negative input needs base case [OK]

Hint: Check base cases cover all input ranges [OK]

Common Mistakes:

• Ignoring negative inputs in recursion
• Assuming fib(-1) is valid without base case
• Changing recursive calls incorrectly

5. You want to write a recursive function to reverse a string. Which base case and recursive case correctly achieve this?

function reverse(str) {
  if (___) {
    return ___;
  }
  return reverse(___) + str[0];
}

hard

A. str.length === 0 / str / str.slice(1)

B. str.length === 1 / str / str.slice(0, -1)

C. str === null / null / str.slice(1)

D. str === '' / str[0] / str.slice(1)

Solution

1. Step 1: Identify base case for empty string

   When string is empty (''), return the empty string to stop recursion.

2. Step 2: Define recursive call correctly

   Call reverse on substring from index 1 to end, then add first character at the end.

3. Final Answer:

   str === '' / str / str.slice(1) -> Option D

4. Quick Check:

   Empty string base case and slice(1) recursive call [OK]

Hint: Base case empty string, recursive call slice(1) [OK]

Common Mistakes:

• Using null instead of empty string as base case
• Returning wrong substring in recursive call
• Incorrect base case condition