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

Time Complexity: Recursion Base Case and Recursive Case

O(n)

Understanding Time Complexity

When using recursion, it is important to know how many times the function calls itself before stopping.

We want to find out how the number of calls grows as the input gets bigger.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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

This code calculates the factorial of a number using recursion with a base case and a recursive case.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The function calls itself once each time with a smaller number.
• How many times: It repeats until the input reaches zero, so about n times.

How Execution Grows With Input

Each time the function calls itself, it does one multiplication and one smaller call.

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

Pattern observation: The number of operations grows directly with the input size.

Final Time Complexity

Time Complexity: O(n)

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

Common Mistake

[X] Wrong: "Recursion always makes the program slow and complex."

[OK] Correct: Simple recursion like this runs in linear time and is easy to understand when base cases stop the calls.

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 recursive case called the function twice 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 stop the recursion and prevent infinite calls

C. To increase the input size for the next call

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 the function returns without calling itself again.

3. Final Answer:

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

4. Quick Check:

   Base case = stop recursion [OK]

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

Common Mistakes:

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

2. Which of the following is the correct syntax for a recursive function in TypeScript that calculates factorial?

easy

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

B. function factorial(n: number): number { while (n > 0) return n * factorial(n - 1); }

C. function factorial(n: number): number { if (n === 1) return 0; else return n + factorial(n - 1); }

D. function factorial(n: number): number { return n * factorial(n + 1); }

Solution

1. Step 1: Check base case correctness

   function factorial(n: number): number { if (n === 0) return 1; else return n * factorial(n - 1); } uses if (n === 0) return 1; which is correct for factorial base case.

2. Step 2: Check recursive call correctness

   function factorial(n: number): number { if (n === 0) return 1; else return n * factorial(n - 1); } calls factorial(n - 1) and multiplies by n, which matches factorial definition.

3. Final Answer:

   function factorial(n: number): number { if (n === 0) return 1; else return n * factorial(n - 1); } -> Option A

4. Quick Check:

   Correct base and recursive case syntax = function factorial(n: number): number { if (n === 0) return 1; else return n * factorial(n - 1); } [OK]

Hint: Base case returns 1 for factorial, recursive calls with n-1 [OK]

Common Mistakes:

• Using while loop instead of recursion
• Wrong base case value
• Recursive call increasing n causing infinite recursion

3. What will be the output of this TypeScript code?

function sumTo(n: number): number {
  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 return values

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

3. Final Answer:

   6 -> Option A

4. Quick Check:

   Sum 1+2+3 = 6 [OK]

Hint: Add n plus sumTo(n-1) until base case 0 [OK]

Common Mistakes:

• Confusing base case return value
• Adding instead of multiplying
• Missing base case causing infinite recursion

4. Identify the error in this recursive function and select the fix.

function countdown(n: number): void {
  if (n === 0) return;
  console.log(n);
  countdown(n + 1);
}
countdown(3);

medium

A. Change console.log(n) to console.log(n - 1)

B. Add a base case for n === 1

C. Remove the base case to allow infinite recursion

D. Change countdown(n + 1) to countdown(n - 1)

Solution

1. Step 1: Analyze recursive call direction

   The function calls countdown(n + 1), which increases n, causing infinite recursion.

2. Step 2: Fix recursive call to decrease n

   Changing to countdown(n - 1) moves towards base case n === 0, stopping recursion.

3. Final Answer:

   Change countdown(n + 1) to countdown(n - 1) -> Option D

4. Quick Check:

   Recursive call must approach base case [OK]

Hint: Recursive calls must reduce input to reach base case [OK]

Common Mistakes:

• Increasing input causing infinite recursion
• Missing or wrong base case
• Changing print statement instead of recursion

5. You want to write a recursive function flatten that takes a nested array of numbers (e.g., [1, [2, [3]], 4]) and returns a flat array [1, 2, 3, 4]. Which approach correctly uses base and recursive cases?

hard

A. Base case: always return input; Recursive case: never call flatten recursively

B. Base case: if input is number, return number; Recursive case: call flatten on input

C. Base case: if input is empty array, return empty array; Recursive case: for each element, if array, call flatten, else add element

D. Base case: if input is array, return input; Recursive case: if input is number, call flatten

Solution

1. Step 1: Define base case for empty array

   If the input array is empty, return an empty array to stop recursion.

2. Step 2: Define recursive case for nested arrays

   For each element, if it is an array, recursively call flatten; if not, add element to result.

3. Final Answer:

   Base case: if input is empty array, return empty array; Recursive case: for each element, if array, call flatten, else add element -> Option C

4. Quick Check:

   Empty array stops recursion, nested arrays flatten recursively [OK]

Hint: Empty array stops recursion; recurse on nested arrays [OK]

Common Mistakes:

• Wrong base case returning number instead of array
• Not checking element type before recursion
• Infinite recursion by missing base case