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๐Ÿ“ŠGraph Theory

๐Ÿ“ŠGraph Theory Unit 5 โ€“ Trees and Forests

Trees and forests are fundamental structures in graph theory, offering a hierarchical way to organize data. They're used in various applications, from computer science algorithms to modeling real-world relationships. These structures have unique properties that make them powerful tools for problem-solving. Understanding trees and forests is crucial for grasping more complex graph concepts and developing efficient algorithms for diverse computational tasks.

What Are Trees?

  • Trees are connected acyclic graphs consisting of nodes (vertices) and edges
  • Every pair of nodes in a tree is connected by exactly one path
  • Trees have a hierarchical structure with a designated root node at the top
  • Each node in a tree has a unique parent node, except for the root which has no parent
  • Nodes with the same parent are called siblings
  • Nodes without children are called leaves or external nodes, while nodes with children are internal nodes
  • The depth of a node is the number of edges from the root to that node
  • The height of a tree is the maximum depth among all nodes in the tree

Tree Properties and Terminology

  • Trees are minimally connected graphs, meaning removing any edge disconnects the graph
  • The number of nodes in a tree is always one more than the number of edges (n=m+1n = m + 1, where nn is the number of nodes and mm is the number of edges)
  • Trees have no cycles, so there is a unique path between any two nodes
  • The degree of a node in a binary tree is at most 2, while in a general tree, there is no restriction on the degree
  • Subtrees are smaller trees within a larger tree, formed by selecting a node as the root and including all its descendants
  • A forest is a collection of disjoint trees
  • The diameter of a tree is the number of edges in the longest path between any two nodes
  • The level of a node is the number of edges on the path from the root to that node

Types of Trees

  • Binary trees have at most two children for each node (left child and right child)
    • Full binary trees have either 0 or 2 children for every node
    • Complete binary trees have all levels filled, except possibly the last level, which is filled from left to right
    • Perfect binary trees have all levels completely filled
  • N-ary trees allow nodes to have more than two children
  • Balanced trees have heights that are logarithmic in the number of nodes (e.g., AVL trees, Red-Black trees)
  • Trie (prefix tree) is a tree-like data structure used for efficient string searching and retrieval
  • Huffman trees are used for data compression by assigning shorter codes to more frequent characters
  • B-trees are self-balancing trees used in databases and file systems to minimize disk accesses

Tree Traversal Algorithms

  • Preorder traversal visits the root, then recursively visits the left subtree, followed by the right subtree
  • Inorder traversal recursively visits the left subtree, then the root, and finally the right subtree
    • Inorder traversal of a binary search tree yields nodes in ascending order
  • Postorder traversal recursively visits the left subtree, then the right subtree, and finally the root
  • Level-order traversal (Breadth-First Search) visits nodes level by level, from left to right
  • Depth-First Search (DFS) explores as far as possible along each branch before backtracking
  • Traversal algorithms can be implemented using recursion or iteration (with the help of a stack or queue)

Forests and Their Characteristics

  • A forest is a disjoint union of trees, i.e., a graph whose connected components are trees
  • Forests can be created by removing edges from a tree or combining multiple trees
  • The number of trees in a forest is equal to the number of connected components in the graph
  • Forests have no cycles, and there is a unique path between any two nodes within the same tree
  • Many tree algorithms can be extended to work on forests by applying them to each tree independently
  • Forests can be used to model hierarchical relationships or represent disjoint sets

Applications of Trees in Graph Theory

  • Trees are used to model hierarchical relationships (e.g., family trees, organizational charts)
  • Spanning trees are subgraphs that include all vertices of a graph and have no cycles
    • Minimum spanning trees (MSTs) have the smallest total edge weight among all spanning trees
  • Trees can represent decision-making processes or game trees (e.g., chess, tic-tac-toe)
  • Prefix trees (tries) are used for efficient string searching and retrieval in text processing
  • Huffman coding uses trees to compress data by assigning shorter codes to more frequent symbols
  • Trees are used in network routing protocols to avoid loops and efficiently propagate information

Problem-Solving with Trees

  • Many graph problems can be solved efficiently using tree algorithms
  • Checking if a graph is a tree can be done by verifying it is connected and has nโˆ’1n-1 edges
  • Finding the shortest path between two nodes in a tree is straightforward since there is only one path
  • Lowest Common Ancestor (LCA) problem finds the deepest node that is an ancestor of two given nodes
  • Diameter of a tree can be found by running two depth-first searches from an arbitrary node
  • Tree isomorphism problem determines if two trees have the same structure, ignoring node labels
  • Counting the number of distinct trees with nn nodes is given by the Catalan numbers: Cn=1n+1(2nn)C_n = \frac{1}{n+1}\binom{2n}{n}

Advanced Tree Concepts

  • Self-balancing trees (e.g., AVL trees, Red-Black trees) maintain logarithmic height through rotations
    • Rotations are local operations that rebalance the tree while preserving the binary search tree property
  • Splay trees are self-adjusting binary search trees that move recently accessed nodes closer to the root
  • Treaps combine the binary search tree property with heap-like priorities for efficient searching and updates
  • Persistent trees allow efficient access to multiple versions of a tree, useful for undo/redo operations
  • Suffix trees represent all suffixes of a string, enabling fast pattern matching and substring queries
  • Tree decomposition and tree-width are used to solve graph problems by exploiting the tree-like structure of graphs
  • Dynamic trees support efficient updates and queries on trees that change over time (e.g., link-cut trees)


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ยฉ 2025 Fiveable Inc. All rights reserved.
APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.