Trees and Binary Trees Overview

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Questions and Answers

What is the characteristic of a tree data structure that distinguishes it from linear data structures?

It must have exactly two children for each node.

It does not store elements in a sequential manner. (correct)

It can only contain integer data types.

It is always sorted in ascending order.

How many edges are present in a tree with 'n' nodes?

n + 1

n - 1 (correct)

What is defined as the depth of a node in a tree?

The maximum number of children any node can have.

The total number of nodes present in the tree.

The length of the path from the root to the node. (correct)

The number of edges from the node to its farthest leaf.

Which of the following best defines a tree's root node?

The topmost node that has no parent. Signup and view all the answers

What type of data structure is a tree specifically categorized as?

Recursive data structure Signup and view all the answers

Which property of trees allows them to have relationships between nodes?

There are no cycles in the tree. Signup and view all the answers

What is the primary purpose of a tree data structure?

To facilitate quick search, insertion, and deletion. Signup and view all the answers

In a tree, what is the term used to describe nodes that have no children?

Leaf nodes Signup and view all the answers

What property defines a full binary tree?

Each node must contain either 0 or 2 children. Signup and view all the answers

How is the maximum number of nodes in a complete binary tree calculated?

2^(h+1) - 1 Signup and view all the answers

Which of the following statements about perfect binary trees is true?

All internal nodes must have 2 children and all leaf nodes at the same level. Signup and view all the answers

What is a characteristic feature of a degenerate binary tree?

All internal nodes have only one child. Signup and view all the answers

In what way is a balanced binary tree defined?

The difference in height between left and right subtrees is at most 1. Signup and view all the answers

What is the minimum height of a complete binary tree with 'n' nodes?

log2(n + 1) - 1 Signup and view all the answers

Which type of binary tree is characterized by nodes added from left to right, filling each level before moving to the next?

Complete Binary Tree Signup and view all the answers

Which traversal technique follows the order: Left, Root, Right?

Inorder Traversal Signup and view all the answers

What is the first step in the search algorithm for a binary search tree (BST)?

Return ROOT if ROOT is NULL Signup and view all the answers

When inserting an item in a BST, what should happen if the item to be inserted is less than the current root's data?

Recursively perform the insert operation in the left subtree Signup and view all the answers

Which of the following actions is performed when deleting a leaf node in a BST?

Free the allocated space and replace the node with NULL Signup and view all the answers

What happens if the node to be deleted from a BST has only one child?

The node is replaced with its child Signup and view all the answers

What must be ensured when inserting into a BST?

The structure must not violate the property of BST Signup and view all the answers

Which situation requires a node with two children to be deleted from the BST?

The node must be replaced in a manner to uphold BST properties Signup and view all the answers

During a search operation in a BST, what happens if the searched item is greater than the root data?

Recursively search the right subtree Signup and view all the answers

What is the result of inserting the first element in an empty BST?

A new root node is created with pointers to NULL Signup and view all the answers

What defines the depth of a node in a tree?

The number of edges from the root node to the node Signup and view all the answers

Which of the following statements is true regarding leaf nodes?

Leaf nodes do not have any child nodes Signup and view all the answers

What is the maximum number of nodes possible in a binary tree of height 3?

15 Signup and view all the answers

How is the height of a binary tree defined?

The number of edges from the root to the deepest leaf node Signup and view all the answers

Which of the following is not a type of tree mentioned?

Circular Tree Signup and view all the answers

In a binary tree, what is the minimum number of nodes possible at height h?

h + 1 Signup and view all the answers

Which of the following best describes 'visiting' in the context of tree traversal?

Checking the value of a node during traversal Signup and view all the answers

Which level corresponds to the child nodes directly connected to the root node?

Level 1 Signup and view all the answers

What is the average code length calculated in the example?

2.52 Signup and view all the answers

How many bits does the total Huffman encoded message consist of in the example provided?

147 Signup and view all the answers

What is a defining characteristic of an AVL tree?

Each node has a balance factor from -1 to 1. Signup and view all the answers

Which action should be taken when a node has been inserted into the right subtree of the left subtree in an AVL tree?

Left-Right Rotation Signup and view all the answers

What does a balance factor of -1 indicate in an AVL tree?

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

What is the first step in the Huffman algorithm as described?

Insert character frequencies into a queue. Signup and view all the answers

In Huffman coding, how many merging operations are performed to create the final tree?

C - 1 Signup and view all the answers

What happens when the balance factor of a node in an AVL tree is outside the range -1 to 1?

Balancing operations are required. Signup and view all the answers

Study Notes

Trees

Trees are hierarchical data structures that store information in a hierarchy style
A tree is a non-linear data structure unlike arrays, linked lists, stack and queue
Data is stored in nodes, linked together by edges
The topmost node is the root node
Each node contains data and links to other nodes, known as children
Trees are recursive as the root node has links to the roots of its subtrees
The number of edges in a tree is n-1, where n is the number of nodes

Binary Tree

A binary tree is a special tree where each node can have at most two children
Each node can have zero, one, or two children

Properties of Binary Tree

At each level i, the maximum number of nodes is 2^i
The height of the tree is the longest path from the root to a leaf node
The maximum number of nodes in a tree with height h is 2^(h+1)-1
The minimum number of nodes in a tree with height h is h+1
If there are n nodes in a binary tree, the minimum height can be calculated as log2(n+1) -1

Types of Binary Tree

Full/Proper/Strict Binary Tree: Each node has 0 or 2 children. The number of leaf nodes is equal to the number of internal nodes plus 1.
Complete Binary Tree: All nodes are completely filled except for the last level, where nodes must be filled as left as possible.
Perfect Binary Tree: All internal nodes have 2 children and all leaf nodes are at the same level.
Degenerate Binary Tree: All internal nodes have only one child.
Balanced Binary Tree: The difference in height between the left and right subtrees is at most 1. AVL and Red-Black Trees are balanced binary trees.

Binary Tree Traversal

Tree traversal involves visiting each node in a specific order.
Inorder Traversal: Left Root Right. Used for searching in BST.
Preorder Traversal: Root Left Right. Used for creating a copy of a tree.
Postorder Traversal: Left Right Root. Used for deleting a tree.

Binary Search Tree (BST)

A binary search tree is a binary tree that follows the property that the value of each node is greater than or equal to all values in its left subtree and less than or equal to all values in its right subtree.
Search in BST: Use a recursive approach. Start at the root node, if the item to be searched is equal to the root node, then return the root node, else if the item is less than the root, search recursively in the left subtree, otherwise search recursively in the right subtree.
Insertion in BST: Add a new node, maintaining the BST properties. Allocate memory for a new node, set its data, let its left and right pointers point to NULL. If the item is less than the root, insert recursively in the left subtree, otherwise in the right subtree.
Deletion in BST: Delete a node while maintaining BST properties.
- Leaf Node: Replace the node with NULL and free the allocated space.
- Node with one child: Replace the node with its child and delete the child.
- Node with two children: Replace the node with its inorder successor (the smallest node in the right subtree).

Huffman Encoding

A greedy algorithm that creates an optimal prefix code, known as Huffman Code
Builds a tree using a set of characters and their frequencies, starting from the leaves and merging nodes in a bottom-up manner
Aims to minimize the average code length for a message

AVL Trees

AVL Trees are height-balanced binary search trees.
Each node has a balance factor, calculated by subtracting the height of its right subtree from the height of its left subtree.
Balance factor is between -1 and 1, allowing the tree to be balanced.
Importance: AVL trees prevent the binary search tree from becoming skewed.

LR Rotations

Used to balance an AVL tree when an insertion causes an imbalance.
Performed when a node is inserted into the right subtree of the left subtree, leading to an unbalanced node.
Involves performing a left rotation followed by a right rotation.

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Description

This quiz covers the fundamental concepts of trees and binary trees in data structures. It discusses the hierarchy of tree nodes, properties of binary trees, and their characteristics, including maximum and minimum nodes and heights. Test your understanding of how trees function as non-linear data structures.