Tree Data Structure Explained

1. Node A node is a fundamental part of a tree that contains data or a value. Each node may point to other nodes, which are called its children. In a tree data structure, the first node is called the root of the tree. 2. Root The root node is the top node of the tree from which a...

1. Node A node is a fundamental part of a tree that contains data or a value. Each node may point to other nodes, which are called its children. In a tree data structure, the first node is called the root of the tree. 2. Root The root node is the top node of the tree from which all other nodes descend. There is exactly one root per tree, and it does not have a parent. 3. Edge An edge is a connection between two nodes in a tree. In a typical tree data structure, each edge connects a parent node to a child node, demonstrating the hierarchical relationship within the tree. 4. Parent and Child A parent node is a node that has one or more nodes connected beneath it, referred to as its children. Conversely, a child node is one of the nodes directly connected to another node above it. 5. Sibling Nodes Sibling nodes are nodes that share the same parent node. They are on the same level of the tree. 6. Leaf (or Terminal) Node A leaf node is a node that does not have any children. It is also known as a terminal node. 7. Internal Node An internal node (or non-terminal node) is any node in the tree that has at least one child. It is not a leaf node.  8. Subtree A subtree is any node in a tree along with its descendants. Each node can be seen as the root of a subtree consisting of its children and all their descendants. 9. Level The level of a node is defined by its distance from the root. The root is at level 0, its direct children are at level 1, and so on. 10. Depth The depth of a node is the number of edges from the node to the tree's root node. Thus, the root node has a depth of 0. 11. Height The height of a node is the number of edges on the longest path from the node to a leaf. The height of a tree is the height of its root node, and the height of a leaf is always 0. 12. Degree (Order) The degree of a node is the number of children that node has. The degree of a tree is the maximum degree of any node in the tree. 13. Path A path in a tree is a sequence of nodes and edges connecting a node with a descendant. Paths are often identified by listing the nodes visited in order from the starting node to the destination node. 14. Forest A forest is a set of n ≥ 0 disjoint trees. When any node and all its descendants are removed from a tree, the remaining collection of disconnected trees is referred to as a forest. Applications File System representation, Organizational Charts BST (Binary Search Tree)  Definition is a binary tree data structure where each node contains a key and satisfies the following properties: the key in any node is greater than the keys stored in its left subtree and less than or equal to the keys stored in its right subtree. Benefits Efficient search operations Dynamic data structure Ordered data Range queries Predecessor and Successor Operations Graphical Representation and Visualization Less Space Overhead compared to hash tables Types 1. Perfect Binary Tree A perfect binary tree is a type of binary tree in which every non-leaf node has exactly two children and all leaf nodes are at the same level. This is a highly ordered form of a binary tree where the total number of nodes doubles at each level. 2. Full Binary Tree A binary tree is full if every node has either 0 or 2 children. This does not require that all levels be completely filled, only that no node has only one child. 3. Complete Binary Tree A binary tree is complete if all levels, except possibly the last, are completely filled, and all nodes in the last level are as left as possible. 4. Balanced Binary Tree A binary tree is generally considered balanced if for any two leaves, the difference in depth is at most one. 5. Degenerate Tree A degenerate tree is a tree where each parent node has only one child. This effectively becomes a linked list, resulting in poor performance for typical binary tree operations, as most operations degenerate to O(n). 6. Skewed Binary Tree A tree is skewed if all nodes are biased towards left or right. This bias results in left-skewed or right-skewed trees. Like degenerate trees, skewed trees tend to have poor performance for standard operations because of their linear structure. Traversal : The three primary types are inorder, preorder, and postorder traversals. Here’s a detailed breakdown of each: Inorder Traversal Process:  1. Visit the left subtree: Recursively perform an inorder traversal on the left child. 2. Visit the node: Process (or "visit") the current node. 3. Visit the right subtree: Recursively perform an inorder traversal on the right child. Key Characteristics:  Sorted Order: In a BST, an inorder traversal outputs the node values in ascending order because all nodes in the left subtree are less than the current node and all nodes in the right subtree are greater.  Use Case: o When Sorted Data is Needed: Ideal when you need to retrieve all elements in sorted order. o Searching in BST: Often used for tasks that require processing elements in order. Example Pseudocode: python Copy def inorder(node): if node is not None: inorder(node.left) print(node.value) # Process the node inorder(node.right) Preorder Traversal Process: 1. Visit the node: Process the current node first. 2. Visit the left subtree: Recursively perform a preorder traversal on the left child. 3. Visit the right subtree: Recursively perform a preorder traversal on the right child. Key Characteristics:  Root First: The root is always visited before its subtrees, which makes it easy to reconstruct the tree from the traversal output.  Use Case: o Tree Copying/Cloning: Ideal for creating a copy of the tree or for serialization because the root is recorded before its children. o Expression Trees: Used to prefix (Polish) notation where operators precede their operands. Example Pseudocode: python Copy def preorder(node): if node is not None: print(node.value) # Process the node preorder(node.left) preorder(node.right) Postorder Traversal Process: 1. Visit the left subtree: Recursively perform a postorder traversal on the left child. 2. Visit the right subtree: Recursively perform a postorder traversal on the right child. 3. Visit the node: Process the current node after visiting its children. Key Characteristics:  Children Before Parent: The node is processed only after its children have been visited. This is especially useful in operations where the children must be processed before the parent (e.g., deletion).  Use Case: o Deleting a Tree: When freeing memory, it’s safe to delete children before deleting the parent. o Postfix Expression Evaluation: Useful in expression trees where operands are evaluated before applying the operator. Example Pseudocode: python Copy def postorder(node): if node is not None: postorder(node.left) postorder(node.right) print(node.value) # Process the node When to Use Each Traversal  Inorder Traversal: o Best for retrieving sorted data: Since it processes nodes in ascending order for BSTs, it's ideal for in-order data processing and search operations that require ordered outputs.  Preorder Traversal:  o Best for tree reconstruction: Because the root node is processed before its children, it preserves the structure of the tree and is useful in scenarios like serialization, cloning, or evaluating prefix expressions.  Postorder Traversal: o Best for deletion and evaluation tasks: Since it processes children before the parent, it’s the go-to method for safely deleting nodes (to avoid accessing freed memory) or evaluating postfix expressions. Removing a node Removing a node from a binary search tree (BST) is handled differently based on the number of children the node has. Here’s an in-depth look at the three main scenarios: 1. Removing a Leaf Node (No Children)  Scenario: The node to be removed does not have any children.  Process: o Simply disconnect the node from its parent. o If the node is the root and has no children, the tree becomes empty.  Example: Imagine a BST where the node with value 20 has no children. You would set the corresponding pointer in its parent (either the left or right child pointer) to null. 2. Removing a Node with One Child  Scenario: The node to be removed has exactly one child (either left or right).  Process: o Bypass the node by linking its parent directly to its single child. o The child takes the place of the node being removed.  Example: If a node with value 30 has only a right child with value 40, you update the parent’s pointer to point directly to the node with value 40, effectively removing the node with value 30 from the tree. 3. Removing a Node with Two Children  Scenario: The node to be removed has two children.  Process: o Find the Inorder Successor or Predecessor:   Inorder Successor: The smallest node in the right subtree (i.e., the leftmost node in the right subtree).  Inorder Predecessor: The largest node in the left subtree (i.e., the rightmost node in the left subtree). o Replace the Node's Value:  Copy the value of the chosen successor or predecessor into the node to be deleted. o Delete the Successor/Predecessor:  Since the successor or predecessor will have at most one child (often a leaf), you can then remove it using either the leaf node or one-child removal process.  #include #include template class Node { public: T data; Node* left; Node* right; Node(T data) : data(data), left(nullptr), right(nullptr) {} }; template class BinarySearchTree { public: Node* root; BinarySearchTree() : root(nullptr) {} // Function to insert a new node with the given data  void insert(T data) { root = insertRec(root, data); } // Function to perform in-order traversal void inOrderTraversal() { std::cout right = insertRec(node->right, data); } return node; } // Utility function for in-order traversal void inOrder(Node* node) { if (node != nullptr) { inOrder(node->left); std::cout data right); } } // Utility function for pre-order traversal void preOrder(Node* node) { if (node != nullptr) { std::cout data left); preOrder(node->right); } } // Utility function for post-order traversal  void postOrder(Node* node) { if (node != nullptr) { postOrder(node->left); postOrder(node->right); std::cout data