17 Questions
Which problem solving tool is mentioned in the text?
Recursion
What is a drawback of recursive methods according to the text?
Inherent inefficiency in some algorithms
Why may a non-recursive method be more efficient than a recursive one?
Overhead associated with many method calls in recursion
In the Fibonacci example provided, what value does the function return if n = 1?
1
What is the formula for computing the nth Fibonacci number using recursion?
$fib(n) = fib(n-1) + fib(n-2)$
What does the function 'height(tree)' return based on the provided recursive definition?
The maximum height between the left and right children of the tree
Why might the original code provided generate a NullPointerException?
Because it doesn't handle cases where one child is null while the other is not
How does the solution provided in choice (1) address the issue of null children?
By handling each child separately if one is null
What purpose do Sentinel Nodes serve in data structures?
To replace null pointers in paths for better performance
What benefits can be gained by using Sentinel Nodes instead of null pointers in a data structure?
Increased speed of operations
How are Sentinel Nodes created in the context presented?
By using a specific constructor 'new SentinelNode(null, null, null)'
What is the base case for determining the height of a binary tree?
When both leftChild and rightChild are null
How is the recursive step for determining the height of a binary tree expressed in terms of its children?
height(tree) = 1 + height(tree.leftChild)
What situation can arise in determining a binary tree's height that requires taking the largest of the sub-trees?
When the rightChild has a greater height than the leftChild
What happens if a binary tree has only one child in terms of determining its height?
The height is calculated based on the child's height
Why is using a FOR loop not natural when finding the height of a binary tree?
Binary trees lack array or list structure for looping through
What attribute of binary tree nodes determines whether they are used for calculating tree height?
isLeaf
Study Notes
Recursion and Data Structures
- Recursion is a powerful problem-solving tool, but it has some drawbacks, making non-recursive (or iterative) methods more efficient in some cases.
- Two reasons why non-recursive methods may be more efficient:
- Overhead associated with a large number of method calls
- Some algorithms are inherently inefficient
- Example of recursive problem: computing the nth Fibonacci number
- Recursive definition of Fibonacci number: fib(n) = 1 if n = 0, 1 if n = 1, fib(n-1) + fib(n-2) if n > 1
Calculating Tree Height
- Recursive definition of tree height: height(tree) = 1 + maximum(height(tree.leftChild), height(tree.rightChild))
- Initial code for calculating tree height:
- public int height() { if ((leftChild == null) && (rightChild == null)) return 0; return 1 + Math.max(leftChild.height(), rightChild.height()); }
- Problem with initial code: NullPointerException when one child is null but not the other
- Two ways to fix the problem:
- Check for the case where one child is null but not the other
- Handle null trees as a base case
- Solution for option 1:
- public int height() { if (leftChild == null) { if (rightChild == null) return 0; else return 1 + rightChild.height(); } else { if (rightChild == null) return 1 + leftChild.height(); else return 1 + Math.max(leftChild.height(), rightChild.height()); } }
Sentinel Nodes
- Definition: A Sentinel Node (or Sentinel) is a node that represents a path terminator, specifically designated and not a data node of the data structure.
- Benefits of using sentinels:
- Increased speed of operations
- Reduced algorithmic code size
- Increased data structure robustness
- How to create a sentinel node: new BinaryTree(null, null, null)
- Using sentinel nodes as an alternative to null as the path terminator
Recursive Functions and Procedures
- When working with recursive data structures, it is natural to develop recursive functions and procedures.
- Example: finding the height of a binary tree
- Base case: the simplest tree, where children are null (i.e., just a root), with a height of 0
- Recursive step: express the height of the tree in terms of the smaller trees (i.e., its children)
- Recursive definition of tree height: height(tree) = 1 + maximum(height(tree.leftChild), height(tree.rightChild))
Explore the differences between recursion and iteration in programming, focusing on the efficiency and drawbacks of each method. Learn about the challenges of recursive methods in comparison to iterative ones through examples like computing the nth Fibonacci number.
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