"A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph."
Tree graphs have a single path between any two vertices. They are often used in the study of hierarchical structures and algorithms.
Graph Theory: This is the mathematical study of networks, graphs, and their properties.
Tree Graphs: These are special kinds of graphs that have no cycles and are connected.
Directed Acyclic Graphs (DAGs): A DAG is a directed acyclic version of a tree graph. It is a directed graph with no directed cycles.
Spanning Trees: A spanning tree is a subset of a connected graph that contains all of the vertices of the graph with minimum number of edges.
Tree Traversal Algorithms: These are algorithms that traverse a tree graph in various ways, such as depth-first or breadth-first.
Binary Trees: These are tree graphs where each node has at most two children.
Balanced Trees: These are tree graphs for which the height difference between the left and right sub-trees of any node is at most one.
Red-Black Trees: These are balanced binary search trees used for efficient searching, insertion, and deletion of nodes in dynamic sets.
AVL Trees: These are balanced binary search trees used for efficient searching, insertion, and deletion of nodes in dynamic sets.
B-Trees: These are multi-level dynamically balanced search trees used for storage that allows operations such as insertion, deletion, and searching with a time complexity of O(log n).
Minimum Spanning Trees: These are subsets of edges in a weighted graph that connect all the vertices with minimum possible total edge cost.
Spanning Forests: These are Union of all spanning trees and acyclic subgraphs of a graph.
Topological Sort: This is a linear ordering of nodes in a directed acyclic graph (DAG) such that every node comes before all the nodes that follow it.
Tree Coloring: This is a method of assigning colors to nodes of a tree graph subject to some constraints.
Huffman Coding: This is a compression technique used to encode messages in a binary tree structure.
Decision Trees: These are tree graphs used in decision analysis to represent decisions, chance events, and outcomes.
Trie Trees: Also known as prefix trees, these are tree data structures that store a set of strings such that multiple prefixes can share the same node.
Merkle Trees: These are tree data structures used in cryptography to verify the integrity and authenticity of data by hashing.
Syntax Trees: These are tree data structures used to represent the syntactic structure of sentences in natural language processing.
Parse Trees: These are tree data structures used in compilers to represent the syntactic structure of a programming language.
Rooted Tree: A tree with a designated root node where each node has a parent-child relationship.
Binary Tree: A rooted tree where each node has at most two children.
Full Binary Tree: A binary tree where every node except the leaves has exactly two children.
Perfect Binary Tree: A binary tree where all leaves are at the same level, and all non-leaf nodes have two children.
Complete Binary Tree: A binary tree where all levels are fully filled except possibly for the last level, which is filled from left to right.
Balanced Binary Tree: A binary tree where the sub-trees of any node differ in height by no more than one.
AVL Tree: A self-balancing binary search tree where the sub-trees of any node differ in height by no more than one.
Red-Black Tree: A self-balancing binary search tree designed to guarantee lower worst-case running time for insertion, deletion, and search operations than many other self-balancing search trees.
B-Tree: A tree data structure that keeps data sorted and allows searches, insertions, and deletions in logarithmic amortized time.
Heap: A binary tree data structure that satisfies the heap property: the key stored in each node is either greater than or equal to or less than or equal to the keys in the node's children.
Trie: A special type of tree data structure used to store associative arrays where the keys are usually strings. Each node represents a single character or a pattern of characters.
Huffman Tree: A special type of binary tree used for data compression, where each leaf node represents a character and the frequency of the character's occurrence determines the shape and position of the node in the tree.
Cartesian Tree: A tree derived from a sequence of numbers where each node represents an element of the sequence and the tree satisfies the binary search tree property.
Tangle Tree: A tree used in computational geometry to store triangulations or perturbations of triangulations.
Spanning Tree: A subgraph of an undirected, connected graph that is a tree which connects all the vertices of the given graph.
Simple Tree: A tree with no repeated nodes or edges, and with no self-loops.
Weighted Tree: A tree where each edge has a weight or cost associated with it.
Directed Tree: A tree where each edge has a direction, and there are no cycles.
Undirected Tree: A tree where each edge is undirected and there are no cycles.
Phylogenetic Tree: A tree data structure used in evolutionary biology to represent the evolutionary relationships among a set of species or other taxa.
"A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees."
"A polytree is a directed acyclic graph (DAG) whose underlying undirected graph is a tree."
"A polyforest is a directed acyclic graph whose underlying undirected graph is a forest."
"The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees."
"A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree—or making all its edges point towards the root—in which case it is called an anti-arborescence or in-tree."
"A rooted tree itself has been defined by some authors as a directed graph."
"A rooted forest is a disjoint union of rooted trees."
"A directed rooted forest is a rooted forest that makes all its edges point away from the root in each rooted tree, in which case it is called a branching or out-forest, or makes all its edges point towards the root in each rooted tree, in which case it is called an anti-branching or in-forest."
"The term tree was coined in 1857 by the British mathematician Arthur Cayley."
"...any two vertices are connected by exactly one path..."
"...any two vertices are connected by at most one path..."
"...the underlying undirected graph is a tree."
"...the underlying undirected graph is a forest."
"A rooted tree may be directed, called a directed rooted tree..."
"...making all its edges point away from the root—in which case it is called an arborescence or out-tree..."
"...making all its edges point towards the root—in which case it is called an anti-arborescence or in-tree."
"A rooted forest may be directed, called a directed rooted forest..."
"...making all its edges point away from the root in each rooted tree—in which case it is called a branching or out-forest..."
"...making all its edges point towards the root in each rooted tree—in which case it is called an anti-branching or in-forest..."