Data Structures Unit 5: Trees

Questions and Answers

Which term refers to the connections between elements in a tree structure?

Leaves

Roots

Nodes

Branches (correct)

What is true about the root node in a tree?

It must have at least one child.

It is the only node without children.

It has no parent. (correct)

It can have multiple parents.

In a binary tree, how many children can each node have at most?

Any number

One

Two (correct)

Three

What type of node does not have any child nodes?

External node Signup and view all the answers

How would you describe a tree used to represent hierarchical file structures?

Inverted tree Signup and view all the answers

What best defines an internal node in a tree?

A node that has children Signup and view all the answers

What distinguishes a binary tree from other types of trees?

Each node has two branches maximum. Signup and view all the answers

What can be said about the presence of a root node in a tree?

A tree can either have a root node or be empty. 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

Which representation of a binary tree is likely to waste memory?

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 is the depth of the root node in a binary tree?

0 Signup and view all the answers

How many parts is each node divided into in a linked representation of a binary tree?

3 Signup and view all the answers

Which of the following sequences is the result of a Preorder traversal on the described binary tree?

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

What does a subtree consist of in a tree T?

The root node and all its descendants Signup and view all the answers

What is a proper subtree?

A subtree that excludes the root node of a tree Signup and view all the answers

What is the primary drawback of sequential storage representation of trees?

High memory consumption for sparse trees Signup and view all the answers

What information is typically stored in the NULL pointers of a Threaded Binary Tree?

Preorder predecessor/successor information Signup and view all the answers

Which traversal method is avoided by using Threaded Binary Trees to improve efficiency?

Recursive traversal Signup and view all the answers

What is a disadvantage of using a Threaded Binary Tree?

It makes the tree structure more complex Signup and view all the answers

In a Binary Search Tree (BST), where can duplicates be found?

In neither subtree, duplicates are not allowed Signup and view all the answers

How is the highest valued element in a Binary Search Tree determined?

By traversing right until no more right child exists Signup and view all the answers

What initial step is taken when trying to search for a key in a Binary Search Tree?

Begin at the root node Signup and view all the answers

What characteristic does a Binary Search Tree maintain regarding its left and right subtrees?

Left subtree has keys less, right subtree has keys greater than the parent key Signup and view all the answers

What is the primary advantage of using Threaded Binary Trees over regular Binary Trees?

They eliminate recursion, reducing memory and time usage Signup and view all the answers

What happens when trying to insert an element into a Binary Search Tree that already contains that element?

The operation is ignored and nothing changes Signup and view all the answers

What is the structure of a node in a Threaded Binary Tree as described in the content?

Includes an additional pointer for threads Signup and view all the answers

What happens to the structure of a Binary Search Tree (BST) if a sorted sequence of values is repeatedly inserted?

It forms a completely skewed BST. Signup and view all the answers

What is the worst-case time complexity for searching or inserting an element in a BST with n nodes?

O(n) Signup and view all the answers

In the insertion algorithm for a BST, when is a new node placed in the left sub-tree?

When it is less than the value in the root node. Signup and view all the answers

How is memory allocated for a new node during the insertion process in a BST?

Using malloc. Signup and view all the answers

When inserting an element into a BST, if the current node is NULL, what is the first action taken?

Create a new node. Signup and view all the answers

What is the appropriate action when the node to be inserted is less than the current node in the insertion process?

Recursively traverse the left sub-tree. Signup and view all the answers

Which of the following correctly describes how to navigate the tree if the new element is greater than the current node during insertion?

Recursively traverse the right sub-tree. Signup and view all the answers

What would the height of a completely skewed BST with n nodes be?

n - 1 Signup and view all the answers

If an element 19 is inserted last in the given sequence, where will it be placed in relation to other existing nodes?

As the right child of 14. Signup and view all the answers

What does the term 'T' represent in the insertion algorithm?

The current node being evaluated. Signup and view all the answers

What is the correct position of the node after inserting 24 into the binary search tree?

Under node 23 Signup and view all the answers

When deleting a node with no children from a binary search tree, what happens to the tree structure?

The node is removed completely. Signup and view all the answers

What type of node is being deleted in the scenario where 14 is removed from the binary search tree?

A node with one child Signup and view all the answers

When deleting a node with two children, which node replaces the deleted node?

The smallest node of the right subtree Signup and view all the answers

What is the first step in deleting node 13 from the binary search tree?

Find the smallest node in the left subtree. Signup and view all the answers

What remains in the binary search tree after deleting node 24?

Node 23 and its children Signup and view all the answers

Which of the following accurately describes a node with two children in a binary search tree?

It must have both left and right children. Signup and view all the answers

What is a potential outcome after deleting a node with one child?

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

What would be the result of deleting the leaf node from the binary search tree structure?

The parent node becomes a leaf node. Signup and view all the answers

Which structure will result after deleting node 19 from the binary search tree?

Node 13 will remain unchanged. Signup and view all the answers

What process occurs after storing the top element from a binary heap into a temporary place?

A new hole is created at the top of the heap. Signup and view all the answers

What happens to cells that are no longer part of the heap after a swap in a binary heap?

They become part of the sorted array. Signup and view all the answers

Which condition must be met for the percolation process to occur in a binary heap?

The element is less than the children of the hole. Signup and view all the answers

How does a degenerate binary tree affect search efficiency?

It results in O(n) efficiency. Signup and view all the answers

What is a characteristic of a complete binary tree?

Nodes are uniformly distributed across subtrees. Signup and view all the answers

What is the maximum height difference allowed for an AVL tree node to maintain its height-balance?

1 Signup and view all the answers

What is the typical height of a binary search tree for an array of n elements?

log2 n + 1 Signup and view all the answers

What algorithms are responsible for maintaining balance in AVL trees?

Insert and remove algorithms. Signup and view all the answers

How do AVL trees differ fundamentally from standard binary search trees?

They maintain a strict height-balance. Signup and view all the answers

In terms of search efficiency, what is the performance of a complete binary tree structure?

O(log2 n) Signup and view all the answers

What is the smallest node in the right sub-tree of the node containing 20?

21 Signup and view all the answers

What property does a max heap satisfy regarding parent and child nodes?

Parent key is greater than or equal to child keys. Signup and view all the answers

Which procedure represents the first step in sorting an array using heapsort?

Build the heap tree. Signup and view all the answers

After deleting the node 20 and replacing it with 21, what is the right child of 21 in the resultant binary tree?

22 Signup and view all the answers

What is the result of performing deleteMax operations on a max heap?

The largest element is deleted. Signup and view all the answers

In the heapsort procedure given an initial array of 6 elements, which of the following represents the highest priority after building the heap tree?

19 Signup and view all the answers

When building a heap from an array, what part of the array is treated as the heap?

A portion considered as the heap. Signup and view all the answers

Which of the following statements is true about the resultant binary tree after deleting 20 and replacing it with 21?

21 has one child. Signup and view all the answers

What should be the next step after building the heap tree if the goal is to sort the array in ascending order?

Perform deleteMax operations. Signup and view all the answers

What is the outcome of the sorting process after deleteMax operations have been performed on the heap?

Elements are sorted in ascending order. Signup and view all the answers

What is the relationship between the values in the subtrees of a node with two keys a1 and a2 in a B-tree?

All values in the middle subtree are greater than a1 and less than a2. Signup and view all the answers

How is the number of child nodes related to the number of keys in a B-tree node?

The number of child nodes is one more than the number of keys stored. Signup and view all the answers

What is the primary advantage of using a B-tree in terms of accessing data?

It optimizes access time by accessing fewer nodes overall. Signup and view all the answers

What characterizes the depth of leaf nodes in a B-tree?

All leaf nodes are kept at the same depth. Signup and view all the answers

In a B+ tree, what additional structure is present at the bottom compared to a B-tree?

An extra level of linked leaves. Signup and view all the answers

What is one critical feature of B+ trees that enhances their performance?

They maintain a very high fanout, often over 100. Signup and view all the answers

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

The information necessary for each child node and storage size constraints. Signup and view all the answers

How does the structure of internal nodes in a 2-3 B-tree differ from that of a standard B-tree?

Internal nodes store either one or two keys. Signup and view all the answers

What is the significance of the balanceFactor in an AVL tree?

It measures the difference in height between the right and left subtrees. Signup and view all the answers

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

The left child replaces the parent node. Signup and view all the answers

What condition triggers a double right rotation in an AVL tree?

The parent is heavy on the left and the left child is heavy on the right. Signup and view all the answers

What is a defining characteristic of B-trees compared to binary search trees?

B-trees allow nodes to have more than two children. Signup and view all the answers

Which of the following best describes the balancing process in AVL trees?

Rotations are used to adjust the positions of nodes. Signup and view all the answers

In a 2-3 B-tree, how many keys are present if an internal node has 3 child nodes?

2 keys Signup and view all the answers

What happens to the number of child nodes in B-trees when data is inserted or removed?

Internal nodes may be joined or split. Signup and view all the answers

Why are AVL trees considered efficient search containers?

Their height does not exceed O(log2 n). Signup and view all the answers

What does a balanceFactor of 0 indicate in an AVL tree?

The left and right subtrees are equal in height. Signup and view all the answers

In terms of performance, why are B-trees favored for databases and filesystems?

They are optimized for large block data access. 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 hierarchical organization of data structures in this quiz focused on trees. Understand how elements are connected and the significance of branches in data representation. Perfect for students studying data structures!