Data Structures: Trees and Their Types

Definition and Types

• A tree is a non-linear hierarchical data structure composed of nodes connected by edges.
• The topmost node is the root.
• Nodes without children are leaves.
• Nodes with children are internal nodes.
• Trees represent hierarchical relationships between elements.
• Different tree types exist, each with unique properties and applications.

Binary Tree

• A binary tree is a tree structure where each node has at most two children: a left child and a right child.
• There's a specific order to elements in the children: Left subtree elements < parent < right subtree elements

Binary Search Tree

• A binary search tree (BST) is a specialized binary tree.
• Each node's left subtree contains values less than the parent node's value, and the right subtree contains values greater than the parent node's value.
• BSTs allow for efficient searching, insertion, and deletion.
• Operations like searching, insertion, and deletion take O(h) time, where h is the height of the tree.

AVL Tree

• An AVL tree is a self-balancing binary search tree.
• It ensures logarithmic time complexity for search, insertion, and deletion.
• It maintains balance by performing rotations to keep the height difference between left and right subtrees of any node at most 1.
• This maintains the tree height close to log n, where n is the number of nodes.

B-Tree

• B-trees are self-balancing tree structures designed for quick data retrieval.
• Commonly used in database indexing due to their speed.
• Compared to binary trees and BSTs, B-trees allow more data per node.
• Nodes can have multiple children.
• The number of keys per node is restricted. This limits the maximum height, affecting search, insert, and delete operations' time complexity.

Red-Black Tree

• A red-black tree is another self-balancing binary search tree.
• It uses color attributes (red or black) to guarantee a logarithmic tree height regarding the number of nodes.
• Insertion and deletion require rebalancing through rotations.
• Red-black trees offer guaranteed logarithmic time complexity for searching, insertion, and deletion.

Applications

• Trees are used in various applications, including:
• Representing file systems
• Implementing dictionaries
• Creating expression parsers
• Building decision trees in machine learning
• Storing hierarchical data (e.g., organizational charts, website structures)
• Game AI (e.g., pathfinding).

Traversal Methods

• Tree traversal methods visit every node:
  • Preorder: Root, left subtree, right subtree (recursive).
  • Inorder: Left subtree, root, right subtree.
  • Postorder: Left subtree, right subtree, root.
  • Level order: Traverse nodes level by level, starting from the root (often implemented with queues).

Time Complexity

• Operation time complexity varies by tree type and height (depth).
• Balanced trees (e.g., AVL trees, red-black trees) have O(height) time complexity for search, insertion, and deletion.
• Unbalanced trees, reaching a degenerate state, can have O(n) complexity for these operations.
• In a balanced tree, height is logarithmic concerning the number of nodes.
• In an unbalanced tree, height can become linear, affecting operation times directly.