AVL Trees and Balance Criteria Quiz

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Questions and Answers

What triggers the need for a tree rotation in an AVL tree?

Updating the value of an existing node

Adding a node that maintains balance

Inserting a node without any impact on balance

Deleting a node that results in an unbalanced state (correct)

Which of the following defines an unbalanced state in a tree?

The height difference between any two subtrees is less than 2

The balance factor is between -1 and 1

The height difference exceeds 1 between the subtrees (correct)

All nodes must have a balance factor of 0

What is the primary action needed after deleting a node from an AVL tree?

Rebalance only the affected subtree and its ancestors (correct)

Leave the tree unchanged if there are no errors

Check and restore balance for the entire tree

Only fix the deleted node

What is the degree of a binary search tree?

2 (A) Signup and view all the answers

How does an M-way search tree differ from a binary search tree?

It can have multiple values per node and more subtrees (D) Signup and view all the answers

What is the first step in the deletion process in a max-heap?

Replacing the root with the last element (A) Signup and view all the answers

What does the down_heapify function primarily ensure?

The max-heap property is maintained (A) Signup and view all the answers

What is the maximum element in a max-heap?

The element at the root node (D) Signup and view all the answers

Which of the following is NOT a type of height-balanced tree?

Max-heap (C) Signup and view all the answers

What operation is performed to extract the maximum element from a max-heap?

Delete the root and reheapify (C) Signup and view all the answers

What mathematical operation is used to determine the left child of a heap node?

2 * parent + 1 (C) Signup and view all the answers

What does the balance factor in an AVL tree signify?

The difference in height between left and right subtrees (B) Signup and view all the answers

What happens if the new root after deletion in a max-heap is smaller than its children?

Down-heapification needs to occur (D) Signup and view all the answers

What is the main disadvantage of a binary search tree (BST)?

It can easily become unbalanced. (B) Signup and view all the answers

What guarantees a balanced tree's search cost to be O(log2n)?

The tree maintains its balance through specific schemes. (A) Signup and view all the answers

In a height-balanced tree, what is the maximum allowed height difference between subtrees of any node?

1 (C) Signup and view all the answers

Which property does a weight-balanced tree guarantee?

The number of inner nodes in the left and right subtrees differ by at most 1. (D) Signup and view all the answers

What characteristic is essential for a tree to be considered height-balanced?

For each node, the heights of its subtrees must differ by at most 1. (A) Signup and view all the answers

What is a potential outcome if a BST remains unbalanced?

Some nodes may become as deep as n. (C) Signup and view all the answers

Why is it important for trees to maintain balance during operations such as insertion?

To optimize the search time to O(log2n). (A) Signup and view all the answers

Which statement best describes the overall goal of balanced trees?

To enable efficient search operations regardless of insertion order. (D) Signup and view all the answers

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

To fix the imbalance caused by inserting a node (D) Signup and view all the answers

What becomes the new root of the tree after performing a left rotation at node A?

Node B (D) Signup and view all the answers

During a right rotation at node C, what happens to node A?

It becomes the left child of node B (C) Signup and view all the answers

After a left rotation, what does node A do with B's left child?

Adopts it as its right child (D) Signup and view all the answers

Which of the following describes a right rotation operation?

Node C becomes the new root (D) Signup and view all the answers

In the context of binary tree rotations, which statement is accurate regarding node ownership?

Node ownership is rearranged during rotations for balancing (D) Signup and view all the answers

What happens to node C's right child during a right rotation at node C?

It becomes the left child of node B (C) Signup and view all the answers

What is the mirror operation of a left rotation in a binary tree?

Right rotation (D) Signup and view all the answers

What is the maximum height difference allowed for a height-balanced tree?

At most 1 (A) Signup and view all the answers

How can the height of a tree be determined during the process of checking if it's height-balanced?

By checking the leaves first and then moving up (A) Signup and view all the answers

What characteristic defines an AVL tree?

A binary search tree with height balance criteria (B) Signup and view all the answers

What happens during the insertion process in an AVL tree?

Re-balancing operations are performed as needed (C) Signup and view all the answers

What is the time complexity for search, insertion, and deletion in an AVL tree?

O(log2n) (B) Signup and view all the answers

What is a perfectly height-balanced tree?

A tree where the left and right subtrees of each node are of equal height (B) Signup and view all the answers

What type of operations are necessary to maintain the properties of a binary search tree when inserting in an AVL tree?

Re-balancing operations while preserving the BST property (C) Signup and view all the answers

How can we check if every node in a tree is height-balanced?

By ensuring that every node meets the height balance criteria as you traverse (C) Signup and view all the answers

What condition indicates that a Left Rotation (Single) is insufficient for balancing a tree?

The right subtree is left heavy. (C) Signup and view all the answers

What is the primary purpose of performing a Right Rotation on a subtree?

To balance the tree when a right-heavy subtree becomes unbalanced. (B) Signup and view all the answers

After inserting node 'B' into a balanced tree, which situation results in an unbalanced state?

The right subtree becomes too heavy on the left side. (A) Signup and view all the answers

In the process of balancing an unbalanced tree, which action must be taken if a single left rotation does not suffice?

Conduct a right rotation on the right child. (D) Signup and view all the answers

What is the visual representation of an unbalanced tree following a left rotation when inserting node 'B'?

A structure with the right subtree being heavier. (A) Signup and view all the answers

What does the term 'negative balance' refer to in the context of tree structures?

The right subtree is heavier on the left side than on the right. (A) Signup and view all the answers

What action should be taken after identifying an unbalanced situation involving the right subtree?

Perform a right rotation on the right subtree. (D) Signup and view all the answers

What strategy does the 'Left-Right Rotation' utilize to correct an unbalanced tree?

Perform a left rotation followed by a right rotation. (C) Signup and view all the answers

Study Notes

Unit 8 - Balanced Trees

Binary Search Trees (BSTs) have a significant drawback: they can easily become unbalanced, making search times as slow as a linked list (O(n)), instead of the ideal O(log₂n).
A balanced tree ensures no leaf node is significantly farther from the root than any other leaf. This optimizes search times.
Height-balanced trees maintain subtree differences of at most 1. Empty subtree height = -1
Weight-balanced trees keep the number of inner nodes in left and right subtrees differing by at most 1.

Module Objectives

Students should be able to describe balanced trees.
Students should be able to define and discuss Heap Trees and their properties.
Students should be able to define and discuss AVL Trees and their properties.
Students should be able to define and discuss B-Trees and their properties.

Course Materials (Binary Search Trees)

BSTs are inefficient if unbalanced. A BST with n nodes can have a depth of n, making search times as slow as a linked list.
Balanced trees significantly improve efficiency, with search times of O(log₂n).

Heap Tree

Heaps are a complete binary tree with two key properties:
- Ordering Property: The value in a node is smaller than all the values in its subtrees (Min-Heap). This is different than the structure of a Binary Search Tree.
- Structural Property: All levels are full, except potentially the bottom level, and the nodes in the bottom level are filled from left to right.

Heap Operations

Heapify: Rearranges the elements in the heap to maintain the heap property.
Find-max (or Find-min): In a max-heap, it returns the maximum element, or in min-heap it returns the smallest.
Insert: Adds a new element to the heap.
Delete: Removes an element from the heap.
Extract Min-Max: Returns and removes the minimum or maximum element from the heap, respectively.

AVL Trees

AVL trees are a type of self-balancing binary search tree.
They maintain balance by ensuring that the height difference between the left and right subtrees of any node is at most 1.
They use rotations (left and right rotations, left-right (double left), and right-left (double right) rotations) to maintain balance during insertions and deletions.

B-Trees

B-trees are a self-balancing tree, designed for disk access.
They store data in a way that minimizes the number of disk accesses during data searches, updates, and deletions.
B-trees have characteristics like being: Perfectly balanced where all leaf nodes are at equivalent heights. Very all the nodes, except the root, have between M/2 and M-1 values of data.
For example, a 3-way tree would have 2 or 3 data values per node and 3 subtrees. A 5-way tree would have 2 to 4 data values and 5 subtrees, etc.
Has special insertion and deletion procedures.

Graph Traversal (BFS and DFS)

Graphs are data structures used to model relationships between objects.
Graphs have Nodes and Edges. Nodes represent the objects and edges represent their relationship.
Graphs can be directed (arrows on the edges to show direction) and undirected.
Weighted graphs assign weights to edges to represent costs or distances.
Graphs can be cyclic or acyclic.
A graph is connected if there is a way to move between any two points in a graph. If not a connected graph, then the graph is unconnected.
Traversal techniques (Breadth-First Search (BFS) and Depth-First Search (DFS)) are used to visit every node and edge in a graph.
BFS visits all the nodes at a given distance from the starting node before moving to the next level.
DFS follows a path as far as possible before backtracking. Marking every node as visited prevents revisiting previously processed nodes.

Hashing and Collisions

Hashing is a technique used to efficiently store and retrieve data from a file or database.
It converts a key value into a relative address (integer) using a special function.
The primary function of hashing is to find or locate a record.
Hashing collisions occur when different keys map to the same relative address.
Many methods exist to effectively deal with collisions during data operation:
- Linear Probing: Searches for the next available empty position after the hash address. This can lead to clustered synonyms and potential inefficiency
- Chaining: Stores synonyms in a linked list at their associated bucket.
- Bucket Addressing: Allocates a group of record positions (the bucket) and stores synonyms sequentially within that bucket.
Load factor (number of key values/number of file positions) is important for assessing efficiency when using hashing schemes to maintain file organizations. Optimal load factors generally fall in the 70%-80% region.

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Description

Test your knowledge on AVL trees and their balancing techniques. This quiz covers key concepts such as tree rotations, unbalanced states, and differences between tree structures. Challenge yourself with questions that delve into the mechanics of AVL trees and their operations.