Fibonacci Using Recursion in DSA Javascript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the base case in a recursive Fibonacci function?

The base case returns the first two Fibonacci numbers directly, usually when n is 0 or 1, to stop the recursion.

beginner

Explain how recursion works in calculating Fibonacci numbers.

Recursion calls the same function with smaller inputs (n-1 and n-2) until it reaches the base case, then combines results to get the final Fibonacci number.

intermediate

What is the time complexity of the naive recursive Fibonacci algorithm?

The time complexity is exponential, approximately O(2^n), because it recalculates the same values many times.

beginner

Show the recursive formula used in Fibonacci calculation.

fib(n) = fib(n-1) + fib(n-2), with fib(0) = 0 and fib(1) = 1 as base cases.

intermediate

Why can recursion be inefficient for Fibonacci calculation without optimization?

Because it repeats calculations for the same inputs multiple times, leading to many redundant calls and slow performance.

What does the base case in a recursive Fibonacci function do?

A Stops recursion by returning a known Fibonacci number

B Calls the function again with the same input

C Calculates Fibonacci using a loop

D Returns the sum of all previous Fibonacci numbers

The base case stops recursion by returning a known Fibonacci number for n=0 or n=1.

Which of these is the correct recursive formula for Fibonacci?

A fib(n) = fib(n-1) + fib(n-2)

B fib(n) = fib(n+1) - fib(n-1)

C fib(n) = fib(n-1) * fib(n-2)

D fib(n) = fib(n-2) - fib(n-3)

The Fibonacci number at n is the sum of the two previous Fibonacci numbers.

What is the main problem with naive recursive Fibonacci calculation?

A It uses too much memory

B It does not return correct results

C It recalculates the same values many times

D It only works for even numbers

Naive recursion recalculates the same Fibonacci numbers repeatedly, causing inefficiency.

What are the base cases for Fibonacci recursion?

A fib(1) = 1 and fib(2) = 2

B fib(0) = 0 and fib(1) = 1

C fib(0) = 1 and fib(1) = 1

D fib(2) = 1 and fib(3) = 2

The base cases are fib(0) = 0 and fib(1) = 1 to stop recursion.

What is the time complexity of naive recursive Fibonacci?

A O(n)

B O(n^2)

C O(log n)

D O(2^n)

Naive recursion has exponential time complexity O(2^n) due to repeated calculations.

Describe how a recursive function calculates the nth Fibonacci number.

Think about breaking the problem into smaller parts.

Explain why naive recursion is inefficient for Fibonacci numbers and how it can be improved.

Consider how storing results can help.

Practice

1. What does a recursive Fibonacci function do when called with fib(4)?

easy

A. Throws a syntax error because recursion is not allowed

B. Returns 4 immediately without any calculation

C. Runs an infinite loop without returning

D. Calculates the 4th Fibonacci number by calling itself with smaller numbers

Solution

1. Step 1: Understand recursive calls for fib(4)

   The function calls fib(3) and fib(2) to calculate fib(4).

2. Step 2: Combine results of smaller calls

   fib(3) and fib(2) are calculated recursively and their sum is returned as fib(4).

3. Final Answer:

   Calculates the 4th Fibonacci number by calling itself with smaller numbers -> Option D

4. Quick Check:

   Recursive calls break problem into smaller parts [OK]

Hint: Recursion breaks problem into smaller calls [OK]

Common Mistakes:

• Thinking recursion returns input directly
• Assuming recursion causes infinite loop always
• Confusing recursion with iteration

2. Which of the following is the correct base case syntax for a recursive Fibonacci function in JavaScript?

easy

A. if (n = 0) return 0;

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

C. if n == 1 then return 1

D. if (n > 1) return n;

Solution

1. Step 1: Identify correct JavaScript syntax for base case

   The base case must check if n is 0 or 1 and return n using proper JavaScript syntax.

2. Step 2: Validate each option

   if (n <= 1) return n; uses correct syntax: if (n <= 1) return n; Options A and B have syntax errors or wrong operators, D is logically incorrect.

3. Final Answer:

   if (n <= 1) return n; -> Option B

4. Quick Check:

   Base case uses if (n <= 1) return n; [OK]

Hint: Base case returns n when n is 0 or 1 [OK]

Common Mistakes:

• Using assignment (=) instead of comparison (== or ===)
• Missing parentheses in if condition
• Incorrect logical condition for base case

3. What is the output of the following code?

function fib(n) {
  if (n <= 1) return n;
  return fib(n - 1) + fib(n - 2);
}
console.log(fib(5));

medium

A. 5

B. 8

C. 3

D. 13

Solution

1. Step 1: Trace fib(5) recursive calls

   fib(5) = fib(4) + fib(3), fib(4) = fib(3) + fib(2), fib(3) = fib(2) + fib(1), and so on until base cases.

2. Step 2: Calculate values bottom-up

   fib(0)=0, fib(1)=1, fib(2)=1, fib(3)=2, fib(4)=3, fib(5)=5.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   Fibonacci(5) = 5 [OK]

Hint: Remember Fibonacci sequence: 0,1,1,2,3,5... [OK]

Common Mistakes:

• Confusing index with Fibonacci value
• Miscounting recursive calls
• Assuming fib(5) is 8 or 13 incorrectly

4. Identify the error in this recursive Fibonacci function:

function fib(n) {
  if (n == 0) return 0;
  if (n == 1) return 1;
  return fib(n - 1) - fib(n - 2);
}

medium

A. The recursive step uses subtraction instead of addition

B. The base cases are incorrect

C. The function will cause infinite recursion

D. The function syntax is invalid

Solution

1. Step 1: Check base cases correctness

   Base cases for n==0 and n==1 correctly return 0 and 1 respectively.

2. Step 2: Analyze recursive step

   The recursive step uses subtraction (fib(n - 1) - fib(n - 2)) instead of addition, which is incorrect for Fibonacci.

3. Final Answer:

   The recursive step uses subtraction instead of addition -> Option A

4. Quick Check:

   Fibonacci sums previous two, not subtracts [OK]

Hint: Fibonacci adds previous two numbers, not subtracts [OK]

Common Mistakes:

• Using wrong operator (+ vs -)
• Confusing base cases
• Assuming syntax error when logic is wrong

5. How can you optimize the recursive Fibonacci function to avoid repeated calculations?

hard

A. Remove base cases to speed up recursion

B. Increase the recursion depth limit

C. Use memoization to store and reuse computed values

D. Replace recursion with a while loop only

Solution

1. Step 1: Understand repeated calculations in recursion

   Recursive Fibonacci recalculates same values many times, causing inefficiency.

2. Step 2: Apply memoization to store results

   Memoization saves results of fib(n) calls in a cache to reuse and avoid repeated work.

3. Final Answer:

   Use memoization to store and reuse computed values -> Option C

4. Quick Check:

   Memoization improves recursion efficiency [OK]

Hint: Store results to avoid repeated recursion calls [OK]

Common Mistakes:

• Thinking increasing recursion limit helps
• Believing loops are only solution
• Removing base cases causing infinite recursion