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Questions and Answers
What is a base case in recursion?
What is a base case in recursion?
How is factorial represented mathematically?
How is factorial represented mathematically?
Which of the following describes the recursive case?
Which of the following describes the recursive case?
What is the value of 0!?
What is the value of 0!?
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In the process of programming, what does the coding phase involve?
In the process of programming, what does the coding phase involve?
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Which of the following statements regarding recursive functions is false?
Which of the following statements regarding recursive functions is false?
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What is a possible output if the input to a factorial function is 3?
What is a possible output if the input to a factorial function is 3?
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What is the first step in the programming process?
What is the first step in the programming process?
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What should happen if the first item in the list is a hashtag?
What should happen if the first item in the list is a hashtag?
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Which statement correctly describes how the recursive case functions?
Which statement correctly describes how the recursive case functions?
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What is the base case in the recursive function 'extract_hashtags'?
What is the base case in the recursive function 'extract_hashtags'?
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When the first item in the list is not a hashtag, what should the function do?
When the first item in the list is not a hashtag, what should the function do?
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Which of the following describes what the final return statement in the function does?
Which of the following describes what the final return statement in the function does?
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What is the base case for the factorial function?
What is the base case for the factorial function?
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How does the recursive case of the factorial function operate?
How does the recursive case of the factorial function operate?
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What will the function return when factorial(0) is called?
What will the function return when factorial(0) is called?
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In the polished version of the factorial function, what condition indicates the recursive case?
In the polished version of the factorial function, what condition indicates the recursive case?
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What is the purpose of including doctests in the function's docstring?
What is the purpose of including doctests in the function's docstring?
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Which of the following is true about the return value of factorial(5)?
Which of the following is true about the return value of factorial(5)?
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What type of value does the factorial function return?
What type of value does the factorial function return?
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Which of the following statements about the recursive factorial function is incorrect?
Which of the following statements about the recursive factorial function is incorrect?
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What is the base case in the recursive factorial implementation?
What is the base case in the recursive factorial implementation?
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Which statement best describes the recursive case for calculating factorial?
Which statement best describes the recursive case for calculating factorial?
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What is the first step in defining a recursive factorial function?
What is the first step in defining a recursive factorial function?
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Why is the initial implementation of the recursive factorial impractical for large numbers?
Why is the initial implementation of the recursive factorial impractical for large numbers?
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In the correct recursive case implementation of factorial, what does 'else' indicate?
In the correct recursive case implementation of factorial, what does 'else' indicate?
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Which of the following correctly describes the factorial of a number n?
Which of the following correctly describes the factorial of a number n?
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What is the primary purpose of using recursion in the factorial function?
What is the primary purpose of using recursion in the factorial function?
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What will occur when n = 0 in the factorial function?
What will occur when n = 0 in the factorial function?
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What is the base condition for the factorial function?
What is the base condition for the factorial function?
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What is the output of the function call factorial(5)?
What is the output of the function call factorial(5)?
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What is the purpose of recursion in the factorial function?
What is the purpose of recursion in the factorial function?
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What would happen if the factorial function is called with a negative number?
What would happen if the factorial function is called with a negative number?
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What pattern does the Fibonacci sequence follow?
What pattern does the Fibonacci sequence follow?
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In the count_hashtags function, what parameter does it take?
In the count_hashtags function, what parameter does it take?
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What would be the result of F(4) in the Fibonacci sequence?
What would be the result of F(4) in the Fibonacci sequence?
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What is a primary use of Python Tutor as suggested in the exercises?
What is a primary use of Python Tutor as suggested in the exercises?
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What is the primary structure of the Fibonacci spiral made up of?
What is the primary structure of the Fibonacci spiral made up of?
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What does the orientation of tiles in the Fibonacci spiral depend on?
What does the orientation of tiles in the Fibonacci spiral depend on?
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What should you check to ensure proper functionality of the tile(...) function?
What should you check to ensure proper functionality of the tile(...) function?
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Why should base cases be ensured in recursive functions?
Why should base cases be ensured in recursive functions?
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Which technique can help optimize recursive implementations for large Fibonacci numbers?
Which technique can help optimize recursive implementations for large Fibonacci numbers?
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Which of the following is NOT a concern with deep recursion in Python?
Which of the following is NOT a concern with deep recursion in Python?
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What is a significant characteristic of the color produced for each tile in the spiral?
What is a significant characteristic of the color produced for each tile in the spiral?
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In the Fibonacci spiral, what is a key reason for ensuring proper implementation of the fib_spiral(...) function?
In the Fibonacci spiral, what is a key reason for ensuring proper implementation of the fib_spiral(...) function?
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What graphical function is suggested to check the implementation of a sample tile?
What graphical function is suggested to check the implementation of a sample tile?
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What is the visual element that forms part of each tile in the Fibonacci spiral?
What is the visual element that forms part of each tile in the Fibonacci spiral?
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Study Notes
Introduction to Programming in Python: Recursion
- Recursion is a function calling itself to solve smaller problem instances.
- It's a technique of divide and conquer, breaking problems into simpler, manageable parts.
- Recursion mimics Russian nesting dolls, with a set of problem instances nested within another.
- Recursion has a finite depth; it must have a base case to stop the chain of calls.
- Main components of recursion:
- Base case: A condition that terminates the recursion.
- Recursive case: The function call itself on a smaller problem.
Factorial
- Factorial, denoted as n!, is the product of all positive integers less than or equal to n.
- Examples:
- 0! = 1 (no positive integers less than 1)
- 1! = 1
- 2! = 2 * 1 = 2
- 3! = 3 * 2 * 1 = 6
- 4! = 4 * 3 * 2 * 1 = 24
Example Recursive Factorial
- Implement a recursive function to compute the factorial of a number.
Programming is a Process (Revisited)
- Analysis: Problem analysis and specification.
- Design: Breaking down problems into smaller sub-problems and formulating algorithms.
- Coding: Implementing the solutions in code.
- Testing: Testing code functionality.
- Debugging: Identifying and fixing bugs.
Example Recursive Factorial: Analysis
- Ensure understanding of the mathematical definition of factorial (only for non-negative integers).
- Determine the requirements:
- Input data is a positive integer.
- Output is an integer (the factorial).
Example Recursive Factorial: Design
- Determine a suitable function name (e.g.,
factorial
). - Define the parameters (e.g.,
n
, for the number). - Identify the base case (e.g., 0! or 1! equal 1).
- Determine the recursive case (e.g., n! = n * (n-1)!).
Example Recursive Factorial: Coding (Incorrect Approach)
- A wrong approach is to manually enumerate the products for each case, an impractical method on large numbers.
- This is a less elegant method of computing factorials recursively.
Example Recursive Factorial: Coding (Correct Approach)
- The recursive case includes:
-
elif n == 2: return 2 * factorial(2 - 1);
-
elif n == 3: return 3 * factorial(3 - 1);
-
Example Recursive Factorial: Coding (Polishing Code)
- Base case:
if n == 0 or n == 1: return 1
- Recursive case:
else: return n * factorial(n-1)
Example Recursive Factorial: Testing (by using doctest
)
- Include a docstring to clearly explain the function's purpose, input parameters, and return type.
- Use
doctests
for testing different input values.
Exercise 14.1: Python Tutor
- Use Python Tutor to visualize the execution of the factorial function.
Exercise 14.1: Python Tutor (Observer)
- Observe how Python keeps track of recursive function calls.
Fibonacci
- A sequence where each number is the sum of the two preceding ones (e.g., 0, 1, 1, 2, 3...).
Exercise 14.2: Recursive Fibonacci
- Implement the Fibonacci sequence recursively.
Exercise 14.3: Counting Hashtags
- Recursively count hashtags (strings starting with '#').
Exercise 14.3: Counting Hashtags (Polished)
- A polished and efficient recursive method for computing hashtags, with optimal considerations for base cases.
Exercise 14.3: Counting Hashtags (Hack)
- A concise recursive method for computing hashtags.
Exercise 14.4: Extracting Hashtags
- Recursively extract hashtags from a list of strings.
Exercise 14.5: Fibonacci Spiral
- Create a spiral visual representation of the Fibonacci sequence using tiles.
Tile Orientation
- Tiles rotate as they are added to the spiral (0°, 90°, 180°, 270°, then back to 0°, etc.)
- The
orient
function handles these rotations.
Best Practices about Recursion
- Infinite Loops: Ensure a proper base case for recursion to terminate.
- Deep Recursion: Be aware that Python has a limit on recursion depth.
- Inefficient Fibonacci Computation: Recursively computing Fibonacci numbers can become very slow for large numbers; use memoization or iterative solutions.
- Inefficient List Operations: Recursive operations on large lists can be much slower than using other Python techniques.
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Description
This quiz explores the concept of recursion in programming using Python. It covers definitions, main components, and the practical application of recursion in calculating factorials. Test your understanding of this essential programming technique!