AVL Search Trees Overview

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Questions and Answers

What is the primary property that distinguishes an AVL tree from a regular binary search tree?

The requirement that the height of the two subtrees differ by at most 1. (correct)

The ability to store more nodes efficiently.

The use of a balanced binary search algorithm.

The allowance of duplicate values in the tree.

What does the balance factor of an AVL node indicate?

The number of nodes in the tree.

The maximum height of the tree's left subtree.

The difference in heights between the node's left and right subtrees. (correct)

The entire height of the tree.

What is the time complexity for searching in an AVL tree with n nodes?

O(n)

O(n log n)

O(1)

O(log n) (correct)

Which of the following is true regarding AVL tree rotations?

Rotations restore balance when the balance factor of a node is -2 or +2. Signup and view all the answers

What happens to the height of an AVL tree after an insertion or deletion?

It requires recalculating and potentially adjusting to rebalance. Signup and view all the answers

Which statement describes an AVL tree's worst-case search time?

It remains O(log n) due to its balancing feature. Signup and view all the answers

In the context of AVL trees, when is a rotation particularly necessary?

When the balance factor of an ancestor changes to -2 or +2. Signup and view all the answers

What is the initial height of a newly instantiated AVL tree?

-1 Signup and view all the answers

What occurs during a single right rotation of a node?

The left child becomes the parent and the node becomes its right child. Signup and view all the answers

In a single left rotation, which of the following statements is true?

The right child becomes the parent and the original node becomes its left child. Signup and view all the answers

What is the purpose of performing a single rotation in a binary tree?

To maintain tree balance after insertions or deletions. Signup and view all the answers

What replaces a node's position during a single right rotation?

The left child of the node. Signup and view all the answers

Which of the following best describes the pivot of a rotation?

It is the deepest unbalanced node in the tree. Signup and view all the answers

What is the main purpose of performing a rotation in a Binary Search Tree (BST)?

To rebalance the tree after insertion or deletion Signup and view all the answers

Which of the following describes the first pivot in a double right-left rotation?

The right child of the deepest unbalanced node Signup and view all the answers

What might happen if a single rotation is performed on a completely unbalanced BST?

The tree will remain unbalanced Signup and view all the answers

After a right rotation is performed, which subtree becomes the left child of the new root?

Left subtree of the original right child Signup and view all the answers

What is the result of executing a left rotation about a node in a BST?

The left child of the node is promoted Signup and view all the answers

Which code segment represents the first step of the rotateLeftRight method?

if( isEmpty()) Signup and view all the answers

In a double left-right rotation, what specifically does the second rotation accomplish?

Repositions the deepest unbalanced node as the new root Signup and view all the answers

How does a rotation affect the BST ordering property?

It has no effect on the ordering property Signup and view all the answers

What is meant by 'deepest unbalanced node' in the context of tree rotations?

A node that causes the tree to exceed allowed balance Signup and view all the answers

What type of rotation is required when a node with a balance factor of -2 has a taller left subtree and the insertion is on the right subtree of its left child?

Left-right rotation Signup and view all the answers

In case 3b of AVL insertions, what is indicated by a balance factor of +2?

The right subtree is taller Signup and view all the answers

Which AVL rotation takes place when a node's right subtree is taller and the insertion is on the left subtree of its right child?

Double right-left rotation Signup and view all the answers

What is the primary purpose of the balance method in the AVLTree insert implementation?

To maintain the balance of the tree Signup and view all the answers

Which statement about the insert operation in a Binary Search Tree (BST) is true?

It compares values to determine the insertion point Signup and view all the answers

What happens to the AVL tree after a deletion that does not cause an imbalance?

No further action is required Signup and view all the answers

How does the insertion method in the AVLTree class differ from that of a Binary Search Tree?

AVLTree manages the balance after insertion Signup and view all the answers

When performing rotations in an AVL tree, what does a balance factor of -2 indicate?

The right subtree is taller than the left subtree Signup and view all the answers

What condition must always be satisfied for a binary search tree (BST) during insertion?

For any node x, T1 < x < T3 must hold true. Signup and view all the answers

Which insertion case in an AVL tree requires a single rotation to rebalance?

Same side (left-left) insertion. Signup and view all the answers

When does an imbalance occur in an AVL tree during insertion?

When a node's balance factor changes from +1 to +2 or -1 to -2. Signup and view all the answers

What action should be taken if an AVL tree insertion causes an imbalance?

Rebalance the tree at the lowest unbalanced ancestor. Signup and view all the answers

In case 2a of an AVL tree insertion, when does a right rotation occur?

The insertion was made in the left subtree of its left child. Signup and view all the answers

Which type of rotation is required to rebalance an AVL tree in case 2b during insertion?

Single left rotation. Signup and view all the answers

What defines an AVL tree's balance factor?

The height of the left subtree minus the height of the right subtree. Signup and view all the answers

What is the purpose of rebalancing an AVL tree after insertion?

To maintain the binary search tree property. Signup and view all the answers

Study Notes

AVL Search Trees

AVL trees are a type of binary search tree with a height-balance property
Each node has a balance factor which determines the difference in height of its left and right subtrees
Subtrees of an AVL tree are also AVL trees
Balance factor is calculated as height(right subtree) - height(left subtree)
AVL tree nodes have a balance factor of -1, 0, or +1
Insertion or deletion in a regular BST can cause large imbalances
Searching in an imbalanced BST has O(n) time complexity
AVL trees are rebalanced after each insertion or deletion
The height of an AVL tree with n nodes is O(log n)
Searching, insertion, and deletion operations on AVL trees take O(log n) time

AVL Tree Implementation

The AVL tree class extends the BinarySearchTree class.
It includes a protected integer variable height initialized to -1.
A method to get the height of the tree
A method to adjust the height of a node after insertion or deletion (calculates new height based on left and right subtree heights)
A method to calculate the balance factor of a node (difference in height between subtrees)

Rotations

A rotation is a process of switching children and parents among two or three adjacent nodes to restore balance to a tree.
Rotations are required after insertions and deletions to maintain the height-balance constraint of the AVL tree
Single rotations (right or left)
Double rotations (right-left or left-right) exist; These require two successive rotations.

Why AVL Trees?

Insertion or deletion on a standard Binary Search Tree can create significant imbalances. This can change the time complexity from O(log n) to O(n).
In the worst-case scenario, searching an unbalanced Binary Search Tree has a time complexity of O(n)
AVL trees perform rebalancing after every insertion or deletion.
Height-balance properties ensure the height of the tree remains O(log n).
This means operations like search, insertion, and deletion have logarithmic time complexity.

Insertion in AVL Trees

Insertion uses a BST insertion algorithm
If an imbalance occurs, rebalancing occurs at the deepest or lowest unbalanced ancestor of the inserted node
Three insertion cases exist:
- Case 1: Insertion doesn't cause an imbalance
- Case 2: Insertion on the same side (left-left or right-right) requiring a single rotation
- Case 3: Insertion on opposite sides (left-right or right-left) requiring a double rotation

Deletion in AVL Trees

Deletion uses a BST deletion algorithm
Rebalancing occurs if an imbalance is detected
Three deletion cases:
- Case 1: Deletion does not cause an imbalance.
- Case 2: Deletion requires a single rotation for rebalance
- Case 3: Deletion requires two or more rotations for rebalance

Additional Notes

The implementation details like methods like rotateRight(), rotateLeft(), rotateLeftRight(), rotateRightLeft()(Code snippets shown) are involved in dynamically adjusting the structure of the AVL tree
AVLTree implementation also includes methods for isEmpty, methods for getting left child, right child subtrees etc, to appropriately adjust the height and balance factors.

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Description

Explore the properties and advantages of AVL trees, a type of balanced binary search tree. This quiz covers key concepts including balance factors, height adjustments, and the performance benefits of AVL trees in terms of searching, insertion, and deletion operations.