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Questions and Answers
What is a defining characteristic of a binary tree?
What is a defining characteristic of a binary tree?
- It consists of at least two disjoint subsets.
- It must have a special node called the root node. (correct)
- It can have any number of children per node.
- All nodes must have two children.
Which property of binary trees indicates the maximum number of nodes at a given level?
Which property of binary trees indicates the maximum number of nodes at a given level?
- A binary tree always contains an equal number of nodes at each level.
- If there are m nodes at level L, there can be at most 2m at level L+1. (correct)
- A binary tree at level L can contain at least L nodes.
- All levels of a binary tree must contain the same number of nodes.
What is a requirement for the left and right subtrees in a binary tree?
What is a requirement for the left and right subtrees in a binary tree?
- They can be empty but must be binary trees themselves. (correct)
- They must contain exactly two nodes each.
- They cannot have nodes that are also roots.
- They must have at least three nodes each.
Which of the following is true about the structure of binary trees?
Which of the following is true about the structure of binary trees?
What defines a leaf node in a binary tree?
What defines a leaf node in a binary tree?
What is a major advantage of using AVL trees?
What is a major advantage of using AVL trees?
What happens to the performance of insert and delete operations in AVL trees due to height balancing?
What happens to the performance of insert and delete operations in AVL trees due to height balancing?
Which statement describes a disadvantage of AVL trees?
Which statement describes a disadvantage of AVL trees?
When might you choose to use splay trees over AVL trees?
When might you choose to use splay trees over AVL trees?
Which of the following correctly characterizes AVL trees?
Which of the following correctly characterizes AVL trees?
What happens when trying to delete from an empty binary search tree (BST)?
What happens when trying to delete from an empty binary search tree (BST)?
In which scenario does a node being the only node in the tree lead to a deletion operation?
In which scenario does a node being the only node in the tree lead to a deletion operation?
What is one of the criteria for AVL tree balance?
What is one of the criteria for AVL tree balance?
Which situation requires specific handling when deleting a node from a BST?
Which situation requires specific handling when deleting a node from a BST?
What is the result of inserting values sequentially in ascending order into a BST?
What is the result of inserting values sequentially in ascending order into a BST?
What must be maintained for a BST to ensure efficient operations?
What must be maintained for a BST to ensure efficient operations?
What condition leads to a deletion error in the context of a BST?
What condition leads to a deletion error in the context of a BST?
How does an AVL tree differ from a traditional BST?
How does an AVL tree differ from a traditional BST?
What is the result of performing a single rotation on a Right-Right (R-R) case?
What is the result of performing a single rotation on a Right-Right (R-R) case?
Which insertion sequence requires a double rotation in a Right-Left (R-L) situation?
Which insertion sequence requires a double rotation in a Right-Left (R-L) situation?
When balancing an AVL tree, how does a Left-Right (L-R) situation resolve?
When balancing an AVL tree, how does a Left-Right (L-R) situation resolve?
For which of the following insertion sequences would a single rotation to the right be required?
For which of the following insertion sequences would a single rotation to the right be required?
Which sequence represents a Left-Right situation that requires balancing?
Which sequence represents a Left-Right situation that requires balancing?
What is the balancing action needed after the insertion of values resulting in a Right-Left case?
What is the balancing action needed after the insertion of values resulting in a Right-Left case?
What happens after a rotation is performed at node x during insertion in an AVL tree?
What happens after a rotation is performed at node x during insertion in an AVL tree?
In a balancing process, which situation requires rotating left to achieve a Left-Left case?
In a balancing process, which situation requires rotating left to achieve a Left-Left case?
What is the output of the inorder traversal of the example binary tree?
What is the output of the inorder traversal of the example binary tree?
In a binary search tree, how is the left subtree of a node defined?
In a binary search tree, how is the left subtree of a node defined?
Which of the following traversal orders involves processing the root node first?
Which of the following traversal orders involves processing the root node first?
Given the preorder traversal of 10, 5, 1, 6, 19, 17, 21, what would be the first node processed in postorder traversal?
Given the preorder traversal of 10, 5, 1, 6, 19, 17, 21, what would be the first node processed in postorder traversal?
Which combination of traversals can uniquely reconstruct a binary tree?
Which combination of traversals can uniquely reconstruct a binary tree?
What would be the output of the postorder traversal for the binary tree in the content?
What would be the output of the postorder traversal for the binary tree in the content?
When processing a binary tree using inorder traversal, which of the following is the correct sequence?
When processing a binary tree using inorder traversal, which of the following is the correct sequence?
If the inorder sequence is 1-5-6-10-17-19-21, which node comes after 10 in the inorder traversal?
If the inorder sequence is 1-5-6-10-17-19-21, which node comes after 10 in the inorder traversal?
In the recursive version of postorder traversal, what is the first operation performed?
In the recursive version of postorder traversal, what is the first operation performed?
Which of the following defines the recursive structure of the preorder traversal in binary trees?
Which of the following defines the recursive structure of the preorder traversal in binary trees?
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Study Notes
Binary Tree Traversals
- Six combinations of tree traversal: LVR, LRV, VLR, VRL, RVL, RLV; focus only on LVR, LRV, VLR.
- Types of traversals with example sequences:
- Inorder: 10, 20, 30
- Preorder: 10, 20, 30
- Postorder: 30, 20, 10
- Preorder example: 50, 34, 12, 5, 23, 20, 77, 56, 98, 79, 120, 100
- Postorder example: 5, 20, 23, 12, 34, 56, 79, 100, 120, 98, 77, 50
- Inorder example: 5, 12, 20, 23, 34, 50, 56, 77, 79, 98, 100, 120
Construction of Binary Trees
- Construct trees using:
- Inorder + Preorder
- Inorder + Postorder
Binary Search Tree (BST)
- A BST is a node-based structure where the left subtree has nodes with lesser keys than the parent node.
- Used in compression algorithms, game trees, and file directory structures.
AVL Trees
- AVL trees name from inventors Adelson, Velski, and Landis; they are self-balancing binary search trees.
- Maintain height-balance condition: for each node, the height difference between left and right subtrees should be -1, 0, or 1.
- Operations include insertion with various rotation types to maintain balance:
- Single rotations (left or right) for imbalances.
- Double rotations for complex situations (e.g., left-right or right-left).
Deletion of Nodes in BST
- Ways to handle deletion of nodes:
- Deleting from an empty tree or a tree containing only one node.
- Deleting a root node.
- Handling nodes with no children, one child, or two children.
Properties of Binary Trees
- If a binary tree has m nodes at level L, it can have at most 2m nodes at level L+1.
- Maximum nodes at level L = 2^L.
- The complexity of operations in a balanced tree can be logarithmic.
Pros and Cons of AVL Trees
- Advantages:
- All operations (insert, find, delete) have logarithmic time complexity due to balanced structure.
- Height balancing ensures constant factor added to operation speed.
- Disadvantages:
- Complex to program and debug despite potential library usage.
- Require additional space for height information.
- Rebalancing can incur time costs, affecting performance.
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