What are Trees?

In computer science, a tree is a widely used data structure that represents a hierarchical structure. It consists of a set of nodes connected by edges. The topmost node in a tree is called the root node, and each node in the tree can have zero or more child nodes.

The tree structure is commonly used to represent hierarchical relationships, such as organization charts, file systems, and family trees.

Properties of Trees

Trees have several important properties:

• Root: The topmost node in a tree.
• Parent: A node that has one or more child nodes.
• Child: A node directly connected to another node when moving away from the root.
• Leaf: A node with no child nodes.
• Height: The number of edges on the longest path from the root to a leaf.
• Depth: The number of edges on the path from a node to the root.

Common Types of Trees

There are various types of trees, each with its own characteristics and applications:

• Binary Tree: A tree in which each node can have at most two children.
• Binary Search Tree: A binary tree in which the left child node is less than the parent node, and the right child node is greater than the parent node.
• AVL Tree: A self-balancing binary search tree with the property that the heights of the two child subtrees of any node differ by at most one.

Understanding the properties and types of trees is essential for efficiently solving problems and implementing algorithms that involve tree structures.

Binary Trees

In computer science, a binary tree is a type of tree data structure in which each node has at most two children: the left child and the right child. The left child node is typically smaller than the parent node, and the right child node is typically larger.

Binary trees have various applications in computer science, including:

• Binary Search Trees: A binary tree that is ordered or sorted, allowing for efficient search, insert, and delete operations.
• Expression Trees: A binary tree that represents an arithmetic expression, with the nodes representing operators and operands.
• Huffman Trees: A binary tree used in data compression, where the nodes represent characters and their frequencies.

Understanding binary trees and their implementation is essential for solving problems and implementing algorithms that involve hierarchical data structures.

Let's start by exploring the basic properties and operations of binary trees in C++:

1#include <iostream>
2using namespace std;
3
4// Definition for a binary tree node
5struct TreeNode {
6  int val;
7  TreeNode* left;
8  TreeNode* right;
9  TreeNode(int x) : val(x), left(NULL), right(NULL) {}
10};
11
12int main() {
13  // Replace this code with your own implementation
14  cout << "Binary Trees" << endl;
15  return 0;
16}

In the code snippet above, we define the basic structure of a binary tree node using a C++ struct. Each node has an integer value and pointers to its left and right children, initialized as NULL.

You can replace the existing code with your own implementation to explore different binary tree operations and algorithms in C++. Happy coding!

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#include <iostream>

using namespace std;

​

int main() {

  // Replace this code with your own implementation

  cout << "Binary Trees" << endl;

  return 0;

}

Tree Traversal

Tree traversal refers to the process of visiting or exploring every node of a tree data structure. It allows us to access the data stored in the tree in a specific order.

There are three main types of tree traversal techniques:

1. Inorder Traversal: In this traversal, we first traverse the left subtree, then visit the root node, and finally traverse the right subtree.
2. Preorder Traversal: In this traversal, we first visit the root node, then traverse the left subtree, and finally traverse the right subtree.
3. Postorder Traversal: In this traversal, we first traverse the left subtree, then traverse the right subtree, and finally visit the root node.

Understanding tree traversal techniques is essential for various algorithms and operations on trees, such as searching, inserting, or deleting nodes.

Let's take a look at an example of performing an inorder traversal on a binary tree in C++:

1replace with code logic

In the code snippet above, we define a binary tree structure using a C++ struct. We then implement the inorderTraversal function to perform the inorder traversal using a stack for iterative traversal. At each node, we replace the existing code with our custom logic.

Feel free to replace the existing code with your own implementation and explore other tree traversal techniques and their applications.

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}

#include <iostream>

#include <stack>

using namespace std;

​

// Definition for a binary tree node

struct TreeNode {

  int val;

  TreeNode* left;

  TreeNode* right;

  TreeNode(int x) : val(x), left(NULL), right(NULL) {}

};

​

// Function to perform inorder traversal

void inorderTraversal(TreeNode* root) {

  if (!root) return;

​

  stack<TreeNode*> s;

  TreeNode* curr = root;

​

  while (curr != NULL || !s.empty()) {

    while (curr != NULL) {

      s.push(curr);

      curr = curr->left;

    }

​

    curr = s.top();

    s.pop();

​

    // Replace this code with your custom logic

Binary Search Trees

Binary Search Trees (BSTs) are a type of tree data structure that provide efficient searching and sorting operations. In a BST, each node has at most two child nodes, referred to as the left child and the right child. The key property of a BST is that the value of each node in the left subtree is less than its parent, and the value of each node in the right subtree is greater than its parent.

BSTs are commonly used to store data in sorted order, making it easy to find, insert, and delete elements from the tree.

Operations on Binary Search Trees

1. Insert: Adding a new element to the tree in a way that maintains the BST property.
2. Search: Finding a particular element in the tree by comparing its value with the values of the nodes.
3. Delete: Removing a node from the tree while maintaining the BST property.

To better understand how BSTs work, let's take a look at an example of implementing a BST in C++.

1#include <iostream>
2using namespace std;
3
4struct Node {
5    int data;
6    Node* left;
7    Node* right;
8
9    Node(int value) {
10        data = value;
11        left = nullptr;
12        right = nullptr;
13    }
14};
15
16// Insert function to add a new node to the BST
17Node* insert(Node* root, int value) {
18    if (root == nullptr) {
19        return new Node(value);
20    }
21
22    if (value < root->data) {
23        root->left = insert(root->left, value);
24    } else if (value > root->data) {
25        root->right = insert(root->right, value);
26    }
27
28    return root;
29}
30
31int main() {
32    // Create an empty BST
33    Node* root = nullptr;
34
35    // Insert nodes into the BST
36    root = insert(root, 50);
37    root = insert(root, 30);
38    root = insert(root, 20);
39    root = insert(root, 40);
40    root = insert(root, 70);
41    root = insert(root, 60);
42    root = insert(root, 80);
43
44    // Display the BST
45    cout << "Binary Search Tree: " << root->data << endl;
46    cout << "               /  \\" << endl;
47    cout << "              /    \\" << endl;
48    cout << "             /      \\" << endl;
49    cout << "            /        \\" << endl;
50    cout << "           /          \\" << endl;
51    cout << "          /            \\" << endl;
52
53    return 0;
54}

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}

#include <iostream>

using namespace std;

​

struct Node {

    int data;

    Node* left;

    Node* right;

​

    Node(int value) {

        data = value;

        left = nullptr;

        right = nullptr;

    }

};

​

// Insert function to add a new node to the BST

Node* insert(Node* root, int value) {

    if (root == nullptr) {

        return new Node(value);

    }

​

    if (value < root->data) {

        root->left = insert(root->left, value);

    } else if (value > root->data) {

        root->right = insert(root->right, value);

    }

​

    return root;

}

AVL Trees

AVL Trees are a type of self-balancing binary search tree, named after the inventors Adelson-Velsky and Landis. In an AVL tree, the heights of the left and right subtrees of any node differ by at most one.

The balancing property of AVL trees ensures that the tree remains balanced, providing efficient searching, insertion, and deletion operations with a time complexity of O(log n).

Balanced Tree

In an AVL tree, every node satisfies the AVL property, which states that the heights of the left and right subtrees of the node differ by at most one. If the difference exceeds one, the tree is considered unbalanced and requires balancing operations to maintain the AVL property.

Balancing Operations

There are four types of balancing operations: left rotation, right rotation, left-right rotation, and right-left rotation. These operations are performed to restore the balance of the tree when it becomes unbalanced due to insertions or deletions.

C++ Code Implementation

Here's an example of AVL tree implementation in C++:

1#include <iostream>
2using namespace std;
3
4struct Node {
5    int data;
6    Node* left;
7    Node* right;
8    int height;
9
10    Node(int value) {
11        data = value;
12        left = nullptr;
13        right = nullptr;
14        height = 1;
15    }
16};
17
18int main() {
19    // AVL tree code implementation
20
21    return 0;
22}

Heap Trees

A heap tree, or simply a heap, is a special type of binary tree that satisfies the heap property:

• For a max heap, for any given node, the value of the node is greater than or equal to the values of its children.
• For a min heap, for any given node, the value of the node is less than or equal to the values of its children.

Priority Queues and Heap Trees

Heap trees are commonly used to implement priority queues, which are abstract data types that allow for efficient insertion and removal of elements with associated priorities.

The max heap is often used to implement a max priority queue, where the element with the highest priority is always at the root of the heap. Similarly, the min heap can be used to implement a min priority queue, where the element with the lowest priority is always at the root of the heap.

Example

Let's take a look at an example of creating a max heap from an array using C++ code:

1#include <iostream>
2#include <vector>
3using namespace std;
4
5// Function to swap two elements
6void swap(int& a, int& b) {
7    int temp = a;
8    a = b;
9    b = temp;
10}
11
12// Function to max heapify the subtree with root at given index
13void maxHeapify(vector<int>& arr, int n, int i) {
14    int largest = i; // Initialize the largest element as root
15    int left = 2 * i + 1; // Left child
16    int right = 2 * i + 2; // Right child
17
18    // If left child is larger than root
19    if (left < n && arr[left] > arr[largest])
20        largest = left;
21
22    // If right child is larger than largest so far
23    if (right < n && arr[right] > arr[largest])
24        largest = right;
25
26    // If largest is not root
27    if (largest != i) {
28        swap(arr[i], arr[largest]);
29
30        // Recursively heapify the affected sub-tree
31        maxHeapify(arr, n, largest);
32    }
33}
34
35// Function to convert an array to max heap
36void buildMaxHeap(vector<int>& arr, int n) {
37    // Index of last non-leaf node
38    int startIdx = (n / 2) - 1;
39
40    // Perform max heapify on each non-leaf node
41    // From right to left
42    for (int i = startIdx; i >= 0; i--) {
43        maxHeapify(arr, n, i);
44    }
45}
46
47// Function to print an array
48void printArray(vector<int>& arr, int n) {
49    for (int i = 0; i < n; i++) {
50        cout << arr[i] << " ";
51    }
52    cout << endl;
53}
54
55int main() {
56    vector<int> arr = { 12, 11, 13, 5, 6, 7 };
57    int n = arr.size();
58
59    cout << "Original array: ";
60    printArray(arr, n);
61
62    buildMaxHeap(arr, n);
63
64    cout << "Max Heap: ";
65    printArray(arr, n);
66
67    return 0;
68}

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}

#include <iostream>

#include <vector>

using namespace std;

​

// Function to swap two elements

void swap(int& a, int& b) {

    int temp = a;

    a = b;

    b = temp;

}

​

// Function to max heapify the subtree with root at given index

void maxHeapify(vector<int>& arr, int n, int i) {

    int largest = i; // Initialize the largest element as root

    int left = 2 * i + 1; // Left child

    int right = 2 * i + 2; // Right child

​

    // If left child is larger than root

    if (left < n && arr[left] > arr[largest])

        largest = left;

​

    // If right child is larger than largest so far

    if (right < n && arr[right] > arr[largest])

        largest = right;

​

    // If largest is not root

    if (largest != i) {

        swap(arr[i], arr[largest]);

​