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Introduction to Trees in Data Structures

In computer science, a tree is a hierarchical data structure that is widely used to model relationships and represent data in a structured form. It is called a “tree” because of its branching structure, resembling an upside-down version of a natural tree.

Key Characteristics of a Tree

  1. Nodes: The fundamental units of a tree. Each node can store data and may have links to other nodes.
    1. Root: The top-most node in the tree. Every tree has one root.
    1. Child: Nodes directly connected and below a node are its children.
    1. Parent: The node directly above a child node is its parent.
    1. Leaf: Nodes without children are called leaf nodes.
  2. Edges: The connections between nodes. An edge connects a parent to its child.
  3. Levels: The depth or level of a node is the number of edges on the path from the root to the node.
  4. Height: The longest path from the root to any leaf node in the tree.
  5. Degree: The number of children a node has.

Tree Properties

  • One root: A tree starts from a single root node.
  • Acyclic: A tree does not contain cycles; there is only one unique path between any two nodes.
  • Connected: All nodes in a tree are connected, meaning there are no isolated nodes.

Types of Trees

  1. Binary Tree: Each node can have at most two children, commonly referred to as the left and right child.
    1. Full Binary Tree: Every node has 0 or 2 children.
    1. Complete Binary Tree: All levels, except possibly the last, are fully filled, and the last level’s nodes are as far left as possible.
    1. Perfect Binary Tree: All internal nodes have two children, and all leaf nodes are at the same level.
  2. Binary Search Tree (BST): A binary tree where the left child contains values smaller than the parent, and the right child contains values larger.
  3. AVL Tree: A self-balancing binary search tree where the height of the two child subtrees of any node differs by at most one.
  4. N-ary Tree: A tree where each node can have up to N children.
  5. Trie: A tree used for storing strings, commonly in dictionaries and text searching.
  6. Heap: A complete binary tree used in priority queues, where each parent node satisfies a specific ordering property (min-heap or max-heap).

Common Applications of Trees

  • Hierarchical Data: Representing file systems, organizational structures, XML/HTML DOM.
  • Searching and Sorting: Binary search trees (BST) and heaps.
  • Networking: Routing tables and multicast communication.
  • AI and Machine Learning: Decision trees and game trees.
  • Data Compression: Huffman encoding uses trees for optimal encoding.

Basic Operations on Trees

  1. Traversal:
    1. In-order: Left → Root → Right
    1. Pre-order: Root → Left → Right
    1. Post-order: Left → Right → Root
    1. Level-order: Traverse each level from top to bottom.
  2. Insertion: Adding a node to the tree following specific rules.
  3. Deletion: Removing a node and restructuring the tree.
  4. Search: Finding a node with a specific value.
  5. Balancing: Ensuring the tree remains balanced (e.g., AVL or Red-Black trees).

Understanding trees is essential because they are foundational in designing efficient algorithms and organizing data hierarchically.