Tree Data Structure Overview

Questions and Answers

What is the main purpose of using a tree data structure?

To store elements in a linear fashion for fast access

To maintain a hierarchy of data (correct)

To allow multiple parent-child relationships

To maximize storage space efficiency

What term is used to describe the connections between elements in a tree?

Edges

Links

Branches (correct)

Connections

Which node is considered the topmost node in a tree structure?

External node

Child node

Leaf node

Root node (correct)

What is an internal node in a tree?

A node that has child nodes Signup and view all the answers

What is true about the leaves of a tree?

They have no child nodes Signup and view all the answers

How many branches does a binary tree have at most for each node?

Two Signup and view all the answers

Which statement is true about the root node in a tree?

It is the only node without a parent Signup and view all the answers

What characterizes a tree that can also be empty?

It may not necessarily have a root node Signup and view all the answers

What does a NULL left pointer indicate in a threaded binary tree during a preorder traversal?

It contains the predecessor information. Signup and view all the answers

What is a significant advantage of using a threaded binary tree compared to regular binary tree traversals?

It reduces memory consumption by avoiding stacks. Signup and view all the answers

Which of the following statements is true for a binary search tree?

The in-order traversal results in a sorted sequence of values. Signup and view all the answers

What data structure is primarily used during traditional binary tree traversals?

Stack Signup and view all the answers

What type of information do NULL pointers store in a threaded binary tree?

Predecessor and successor information. Signup and view all the answers

How can the highest valued element in a binary search tree be identified?

By traversing right from the root until a node with no right link is found. Signup and view all the answers

What can be a disadvantage of threaded binary trees compared to regular binary trees?

They make the tree structure more complex. Signup and view all the answers

What must be true about the subtrees of a binary search tree?

They must maintain the binary search property. Signup and view all the answers

What occurs during the insertion operation in a binary search tree if the key is already present?

The key is ignored. Signup and view all the answers

What is the height of a leaf node in a binary tree?

0 Signup and view all the answers

In the context of binary trees, how is the depth of a node defined?

Length of the path from the node to the root Signup and view all the answers

Which storage representation of a binary tree is prone to memory wastage?

Sequential storage representation Signup and view all the answers

In which traversal method is the root processed last?

Postorder Signup and view all the answers

What does a proper subtree correspond to in terms of a binary tree?

A subtree excluding the root of the main tree Signup and view all the answers

Which of the following represents the correct pre-order traversal output of a binary tree?

7, 1, 0, 3, 2, 5, 4, 6, 9, 8, 10 Signup and view all the answers

What does each node in the linked representation of a binary tree consist of?

Value and pointers to left and right child Signup and view all the answers

What is the definition of a subtree in relation to a binary tree?

A tree made up of a node and all its descendants Signup and view all the answers

In inorder traversal, which step is taken immediately after visiting the root?

Traverse the right subtree Signup and view all the answers

What is the result of inserting the value 24 into the binary search tree mentioned?

24 becomes the left child of 19. Signup and view all the answers

Which case applies when deleting a leaf node from a binary search tree?

The node is removed without affecting its parent. Signup and view all the answers

What happens to the binary search tree structure when deleting a node with one child?

The child of the deleted node takes its place. Signup and view all the answers

In the deletion case where a node has two children, what should be done?

Replace it with the smallest node in the right subtree. Signup and view all the answers

After deleting the value 14 from the tree, which node becomes a direct child of 13?

9 remains unchanged. Signup and view all the answers

When deleting the node 13, why is 14 used to replace it?

14 is the smallest node in the right subtree. Signup and view all the answers

What is the result of deleting the maximum element 10 from the heap?

10 is replaced by the last element of the heap and the heap is then adjusted. Signup and view all the answers

What defines a leaf node in a binary search tree?

A node with no children. Signup and view all the answers

Which of the following describes the deletion process when a node has two children?

The node is replaced with its in-order successor. Signup and view all the answers

How is height-balance defined in AVL trees compared to regular binary search trees?

For AVL trees, the difference in height of subtrees is at most 1. Signup and view all the answers

What structure is formed after deleting the node 24 from the binary search tree?

The structure remains unchanged. Signup and view all the answers

What happens during the percolation step in a binary heap?

An element is swapped down until it finds its correct position. Signup and view all the answers

Which situation describes a degenerate binary search tree?

Most elements form a linked list in one of its subtrees. Signup and view all the answers

How does deleting a node from a binary search tree affect the in-order traversal?

It does not affect in-order traversal. Signup and view all the answers

Why are AVL trees considered balanced compared to binary search trees?

The structure of AVL trees maintains a minimal height during operations. Signup and view all the answers

What occurs when an element is swapped with the last element in the heap?

The former last element is no longer part of the heap. Signup and view all the answers

What is a key characteristic of a complete binary tree?

It must have all nodes filled from top to bottom left to right. Signup and view all the answers

In the context of binary heaps, what does percolating down achieve?

It ensures the heap property is kept after an element is swapped. Signup and view all the answers

What defines the height of an AVL tree in relation to a binary search tree?

Binary search trees can have an arbitrary height depending on the order of inserts. Signup and view all the answers

What will happen to the elements during a DeleteMax operation in a binary heap?

The maximum element is removed and the last element takes its place. Signup and view all the answers

What is the correct method for replacing the node containing 20 in the binary search tree?

Replace it with the smallest node from the right subtree. Signup and view all the answers

What characteristic defines a max heap?

The highest key is always in the root node. Signup and view all the answers

Which of the following is a valid step in the heapsort process?

Perform deleteMin operations to form a sorted array. Signup and view all the answers

When building a heap tree, which part of the array is considered the heap?

A portion that will be referred to as the heap while the rest is original. Signup and view all the answers

What is the result of performing the deleteMax operation in a heapsort?

It stores the deleted element in the sorted array. Signup and view all the answers

Which value should be moved in order for the node with value 15 to maintain the max heap property?

19 should be moved above 15. Signup and view all the answers

What happens when the smallest node in a subtree is used to replace a deleted node in a binary search tree?

The deleted node's children are reassigned. Signup and view all the answers

In heapsort, why is the heap tree built in descending order?

To prioritize larger elements for deletion. Signup and view all the answers

After the node containing 20 is replaced, what remains as the left child of 21?

Node 14. Signup and view all the answers

What must be done after performing the deleteMax operation in heapsort?

Percolate down to restore the heap property. Signup and view all the answers

What is the height of a completely skewed Binary Search Tree (BST) with n nodes?

n - 1 Signup and view all the answers

Which of the following best describes the worst-case complexity of searching for an element in a BST with n nodes?

O(n) Signup and view all the answers

What must be true for a new node to be added to the left sub-tree during insertion into a BST?

The new node's value is less than the root's value. Signup and view all the answers

If the root of a BST is NULL, what is the first step in inserting a new node?

Create the root node. Signup and view all the answers

What procedure is executed if the new node's data is less than the data in the current node during insertion?

Traverse the left sub-tree recursively. Signup and view all the answers

In the provided algorithm, what does the line 'T->right = insert(data, T->right);' indicate?

Recursion on the right sub-tree. Signup and view all the answers

After inserting nodes 10, 20, and 30 into a BST, how will the structure look like?

10 as root, 20 as right child, 30 as right child of 20. Signup and view all the answers

How is memory allocated for a new node in the insertion algorithm?

Using malloc. Signup and view all the answers

If you insert 23 into an empty BST after 20, what will the parent-child relationship be?

23 is the right child of 20. Signup and view all the answers

What is a significant defect of repeatedly inserting a sorted sequence into a BST?

Forms a completely skewed tree. Signup and view all the answers

What is a primary characteristic of a 2-3 B-tree regarding its internal nodes?

They can store either one or two keys. Signup and view all the answers

How does a B-tree maintain balance?

By requiring all leaf nodes to be at the same depth. Signup and view all the answers

What is the significance of a B+ tree having a high fanout?

It reduces the number of I/O operations required to find an element. Signup and view all the answers

Which statement correctly describes leaf nodes in a B+ tree?

They are linked to facilitate efficient traversal. Signup and view all the answers

What happens when the overall depth of a B-tree increases?

All leaf nodes are one node farther from the root. Signup and view all the answers

In what context are B-trees particularly advantageous?

When the time to access node data exceeds processing time. Signup and view all the answers

What is the relationship between the number of keys and child nodes in a B-tree node?

The number of child nodes exceeds the number of keys by one. Signup and view all the answers

What defines the maximum number of child nodes in a B-tree?

The size of a full disk block or equivalent storage size. Signup and view all the answers

What is the maximum height of an AVL tree in terms of the number of nodes n?

O(log2 n) Signup and view all the answers

What indicates that an AVL tree node is 'heavy on the left'?

balanceFactor < 0 Signup and view all the answers

What occurs during a single right rotation in an AVL tree?

The left child of a node becomes the new root of the subtree. Signup and view all the answers

In a 2-3 B-tree, how many child nodes can an internal node have?

2 or 3 Signup and view all the answers

What is the primary reason B-trees are preferred for systems that read and write large blocks of data?

They require less frequent re-balancing. Signup and view all the answers

What does the balanceFactor of a balanced AVL tree node equal?

0 Signup and view all the answers

Which operation is necessary to maintain the balance of an AVL tree?

Node rotation Signup and view all the answers

What type of rotation occurs when a parent node is heavy on the left and its left child is also heavy on the right?

Double right rotation Signup and view all the answers

In B-trees, what do keys in internal nodes represent?

Separation values dividing subtrees Signup and view all the answers

When does a node’s balanceFactor indicate it is 'heavy on the right'?

balanceFactor = +1 Signup and view all the answers

Study Notes

Tree Data Structure

A tree is a non-linear data structure organized hierarchically, useful for maintaining data hierarchy.
Each tree element (node) can connect to multiple child nodes, forming branches.
The top node is known as the root; leaves are nodes without children.
Binary trees are a specialized type where each node has a maximum of two child nodes.

Terminology

Node: Contains a value or can represent a sub-tree.
Child Node: A node directly connected beneath another node (parent).
Parent Node: A node that has one or more child nodes.
Internal Node: Any node with child nodes.
External Node (Leaf): Any node without child nodes.
Height of Node: Length of the longest downward path to a leaf.
Depth of Node: Length of the path from the root to the node.
Subtree: A tree consisting of a node and all its descendants.

Tree Representation

C Declaration:

typedef struct TREE {
    int data;
    struct TREE *left;
    struct TREE *right;
} TREE;

Sequential Storage Representation: Uses arrays but can waste memory; best for complete or full binary trees.
Linked Storage Representation: Uses pointers to represent child nodes directly; more efficient but can involve NULL pointers.

Binary Tree Traversal

Preorder: Visit root, traverse left subtree, then right subtree.
Inorder: Traverse left subtree, visit root, then traverse right subtree.
Postorder: Traverse left subtree, then right subtree, and visit the root.
Preorder output example: 7, 1, 0, 3, 2, 5, 4, 6, 9, 8, 10
Inorder output example: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Postorder output example: 0, 2, 4, 6, 5, 3, 1, 8, 10, 9, 7

Issues with Binary Tree Traversal

High memory usage due to stacks/queues for tracking positions.
Many NULL pointers increase space inefficiency.
Difficulty in finding successor nodes.

Threaded Binary Trees

Designed to alleviate traversal issues by using NULL pointers to store predecessor/successor information.
Threaded pointers provide a way to traverse the tree without stacking nodes.
Node structure includes a pointer for threading.

Binary Search Tree (BST)

A node-based tree where:
- Left subtree contains nodes with lesser keys.
- Right subtree contains nodes with greater keys.
- No duplicating nodes allowed.
Inorder traversal will yield sorted node values.
Search and insertion operations leverage the BST properties for efficiency.

Insertion and Deletion in BST

Insertion:
- If the target node is NULL, a new node is created.
- Elements less than the current node go to the left; greater go to the right.
Deletion Cases:
- No children: Simply remove the node.
- One child: Remove the node and link its parent directly to its child.
- Two children: Replace the node with its in-order successor or predecessor, then delete that successor/predecessor.

Heap and Heap Sort

Heaps are specialized tree structures that satisfy the heap property:
- Max Heap: Parent nodes are greater than or equal to child nodes.
- Min Heap: Parent nodes are less than or equal to child nodes.
Heap Sort:
- Build a heap tree using the given elements.
- Perform deleteMin operations to sort the array in ascending order.### Heap Trees and Sorting
To achieve an ascending order of elements, a heap tree is constructed in descending order, prioritizing the largest element.
A single array is used: part represents the heap, while the remainder represents the original or sorted array.
Array segment color coding: white for the original array, blue for the heap, red for the sorted array.
Example starting array: 15, 19, 10, 7, 17, 6.

Building the Heap Tree

The heap is built by starting from the rightmost node at height 1 and percolating down as needed.
If a node's children are smaller, no action is taken; otherwise, swaps occur to maintain the heap property.
The process ends when all necessary nodes are adjusted, ensuring the largest elements are positioned correctly.

Sorting with DeleteMax Operations

The DeleteMax operation removes the top element (the maximum) and repositions elements accordingly.
Elements are stored temporarily during operations, and holes are created at the top as elements are swapped.
The maximum is always replaced from the heap, shifting down elements and maintaining the heap condition until the array is fully sorted.

AVL Trees

AVL trees are a type of binary search tree designed to remain balanced for efficient data access.
They prevent degeneracy where trees become skewed, maintaining height-balance between left and right subtrees.
The height of an AVL tree is O(log2 n), optimizing search efficiency compared to unbalanced trees.

AVL Tree Nodes and Balancing

Each AVL tree node contains a balance factor measuring the difference between subtree heights.
Balance factors indicate whether a node is left-heavy, right-heavy, or balanced.
Rotations are used to restore balance when nodes become unbalanced due to insertions or deletions.

Rotations to Maintain Balance

Single right rotation occurs when the parent and left child both become left-heavy.
A double right rotation is applied when the parent is left-heavy, and the left child is right-heavy.
These rotations adjust the relationships among nodes to preserve the AVL tree properties.

B-Trees

A B-tree is a data structure that maintains sorted data for logarithmic time access, insert, and delete operations.
Internal nodes can have multiple children, allowing for efficient handling of large data blocks.
B-trees maintain balance by keeping all leaf nodes at the same depth.

B+ Trees

A B+ tree is a variation of a B-tree where only keys are stored, with a linked list of leaves added at the bottom.
They feature high fanout, reducing I/O operations and improving performance in block-oriented contexts like filesystems.
Ideal for storing large volumes of data, B+ trees allow quick access and efficient retrieval in storage systems.

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Description

Explore the fundamental concepts of tree data structures through this quiz. Learn about nodes, child nodes, roots, and binary trees, and understand their hierarchical organization. Whether you're a beginner or looking to refresh your knowledge, this quiz will enhance your understanding of tree structures.