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?

<p>Rotations restore balance when the balance factor of a node is -2 or +2. (B)</p> Signup and view all the answers

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

<p>It requires recalculating and potentially adjusting to rebalance. (B)</p> Signup and view all the answers

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

<p>It remains O(log n) due to its balancing feature. (C)</p> Signup and view all the answers

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

<p>When the balance factor of an ancestor changes to -2 or +2. (C)</p> Signup and view all the answers

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

<p>-1 (B)</p> Signup and view all the answers

What occurs during a single right rotation of a node?

<p>The left child becomes the parent and the node becomes its right child. (B)</p> Signup and view all the answers

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

<p>The right child becomes the parent and the original node becomes its left child. (B)</p> Signup and view all the answers

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

<p>To maintain tree balance after insertions or deletions. (A)</p> Signup and view all the answers

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

<p>The left child of the node. (D)</p> Signup and view all the answers

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

<p>It is the deepest unbalanced node in the tree. (B)</p> Signup and view all the answers

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

<p>To rebalance the tree after insertion or deletion (D)</p> Signup and view all the answers

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

<p>The right child of the deepest unbalanced node (C)</p> Signup and view all the answers

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

<p>The tree will remain unbalanced (B)</p> Signup and view all the answers

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

<p>Left subtree of the original right child (C)</p> Signup and view all the answers

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

<p>The left child of the node is promoted (C)</p> Signup and view all the answers

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

<p>if( isEmpty()) (A)</p> Signup and view all the answers

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

<p>Repositions the deepest unbalanced node as the new root (C)</p> Signup and view all the answers

How does a rotation affect the BST ordering property?

<p>It has no effect on the ordering property (D)</p> Signup and view all the answers

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

<p>A node that causes the tree to exceed allowed balance (C)</p> 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?

<p>Left-right rotation (D)</p> Signup and view all the answers

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

<p>The right subtree is taller (C)</p> 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?

<p>Double right-left rotation (B)</p> Signup and view all the answers

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

<p>To maintain the balance of the tree (B)</p> Signup and view all the answers

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

<p>It compares values to determine the insertion point (B)</p> Signup and view all the answers

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

<p>No further action is required (B)</p> Signup and view all the answers

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

<p>AVLTree manages the balance after insertion (C)</p> Signup and view all the answers

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

<p>The right subtree is taller than the left subtree (D)</p> Signup and view all the answers

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

<p>For any node x, T1 &lt; x &lt; T3 must hold true. (B)</p> Signup and view all the answers

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

<p>Same side (left-left) insertion. (A)</p> Signup and view all the answers

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

<p>When a node's balance factor changes from +1 to +2 or -1 to -2. (C)</p> Signup and view all the answers

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

<p>Rebalance the tree at the lowest unbalanced ancestor. (C)</p> Signup and view all the answers

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

<p>The insertion was made in the left subtree of its left child. (A)</p> Signup and view all the answers

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

<p>Single left rotation. (C)</p> Signup and view all the answers

What defines an AVL tree's balance factor?

<p>The height of the left subtree minus the height of the right subtree. (D)</p> Signup and view all the answers

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

<p>To maintain the binary search tree property. (C)</p> Signup and view all the answers

Flashcards

What is an AVL Tree?

A binary search tree that maintains a height balance property where the heights of subtrees differ by at most 1. This ensures that operations remain efficient even with insertions and deletions.

Balance Factor

The difference between the heights of the right and left subtrees of a node in an AVL tree. It can be -1, 0, or +1.

Rotation

A process of rearranging nodes in an AVL tree to restore balance after an insertion or deletion. It involves switching children and parents between two or three adjacent nodes.

Why AVL Trees?

Insertions or deletions in an ordinary binary search tree can lead to large imbalances, affecting search efficiency. AVL trees, being balanced, ensure that searches, insertions, and deletions remain efficient with O(log n) time complexity.

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AVL Subtrees Property

A subtree of an AVL tree is also an AVL tree, inheriting its balancing property.

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Deepest Unbalanced Node

The deepest node in an AVL tree whose balance factor has become -2 or +2 after an insertion or deletion needs to be rotated to restore balance.

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Single Right Rotation

A type of tree rotation in AVL trees where the left child of a node becomes the parent, the node becomes the right child of the former left child, and the right child of the former left child becomes the left child of the node.

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Single Left Rotation

A type of tree rotation in AVL trees where the right child of a node becomes the parent, the node becomes the left child of the former right child, and the left child of the former right child becomes the right child of the node.

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Double Right-Left Rotation

A right rotation followed by a left rotation in an AVL tree.

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Double Left-Right Rotation

A left rotation followed by a right rotation in an AVL tree.

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Right Rotation (Case 2a)

A rotation where the pivot node is the right child of the lowest unbalanced ancestor (with a balance factor of -2) and the insertion was on the left subtree of the pivot's left child. This operation involves rotating the pivot node to the right.

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Left Rotation (Case 2b)

A rotation where the pivot node is the left child of the lowest unbalanced ancestor (with a balance factor of +2) and the insertion was on the right subtree of the pivot's right child. This operation involves rotating the pivot node to the left.

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Same Side Insertion (Left-Left)

This occurs when the new node is inserted into the left subtree of the left child of the lowest unbalanced ancestor (with a balance factor of -2). This requires a single right rotation to restore balance.

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Same Side Insertion (Right-Right)

This occurs when the new node is inserted into the right subtree of the right child of the lowest unbalanced ancestor (with a balance factor of +2). This also requires a single left rotation to rebalance.

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Opposite Side Insertion (Left-Right or Right-Left)

This occurs when the new node is inserted into the right subtree of the left child of the lowest unbalanced ancestor (with a balance factor of -2) or into the left subtree of the right child of the lowest unbalanced ancestor (with a balance factor of +2). This requires a double rotation to rebalance.

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Left Rotation

A single rotation where the deepest unbalanced node is the left child and its right child becomes the new parent.

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Right Rotation

A single rotation where the deepest unbalanced node is the right child and its left child becomes the new parent.

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AVL Property

Ensures that searches, insertions, and deletions remain efficient in AVL trees by maintaining a height balance property where the heights of subtrees differ by at most 1. This ensures that operations remain efficient even with insertions and deletions.

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Rebalancing

The process of restoring balance after an insertion or deletion in an AVL tree. It involves determining the deepest unbalanced node and performing the appropriate rotations.

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Adjusting Height

The process of updating the height of a node after a rotation in an AVL tree. This ensures that the balance factors of the nodes are correctly calculated after a structure change.

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AVL Tree

A type of binary search tree that keeps its balance by ensuring the heights of its subtrees differ by at most 1.

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Double Rotation

A combination of two rotations (left and right) in a specific sequence to restore balance.

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Right-Left Rotation

A double rotation where we first rotate to the right and then to the left.

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Left-Right Rotation

A double rotation where we first rotate to the left and then to the right.

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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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