--- title: Trees description: Trees are connected, acyclic undirected graphs. Learners will understand the structural properties of trees, spanning tree algorithms, and how to use these structures for hierarchical data representation and efficient searching. category: mathematics subcategory: graph-theory difficulty: beginner, intermediate, advanced url: /subject/trees --- # Trees Trees are connected, acyclic undirected graphs. Learners will understand the structural properties of trees, spanning tree algorithms, and how to use these structures for hierarchical data representation and efficient searching. ## Available Resources 4 Courses • 4 Websites ## Websites ### 1. Art of Problem Solving - Trees A short Art of Problem Solving wiki entry on trees in graph theory, stating the definition of a tree as a connected acyclic graph, the n-1 edge property, forests and rooted trees, aimed at students preparing for math competitions. **Difficulty:** Beginner | **Price:** Free **Link:** https://artofproblemsolving.com/wiki/index.php/Trees **Tags:** graph-theory, trees, competition-math, combinatorics ### 2. Tree (graph theory) - Wikipedia Wikipedia's reference article on trees in graph theory, covering equivalent definitions, forests, rooted and ordered trees, spanning trees, and enumeration results such as Cayley's formula. Useful for checking definitions and finding links to related concepts and primary sources. **Difficulty:** Beginner | **Price:** Free **Link:** https://en.wikipedia.org/wiki/Tree_(graph_theory) **Tags:** graph-theory, trees, spanning-trees, cayleys-formula ### 3. Diestel Graph Theory Official site for Reinhard Diestel's Springer graduate text, where the main text is readable free online and paid eBook editions add the full apparatus. It covers matching, connectivity, planarity, colouring, flows, extremal theory, and minors. **Difficulty:** Beginner | **Price:** Free **Link:** https://diestel-graph-theory.com/ **Tags:** graph-theory, textbook, discrete-mathematics, combinatorics, free-online ### 4. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Graph Theory Learn graph theory focusing on trees with Dr. Sarada Herke! Explore fundamentals and applications in this comprehensive course. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLoJC20gNfC2gmT_5WgwYwGMvgCjYVsIQg **Tags:** graph-theory, trees, discrete-mathematics, spanning-trees ### 2. Mathematics for Computer Science (MIT 6.042J) **Author:** Tom Leighton, Marten van Dijk Discrete mathematics for computer science with an emphasis on definitions and proofs: logic, induction, sets and relations, graph theory, modular arithmetic, asymptotics, counting and discrete probability. 25 lecture videos, problem sets and exams with solutions build fluency in writing proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2010/ **Tags:** discrete-mathematics, mathematical-proofs, induction, graph-theory, combinatorics, discrete-probability ### 3. Algorithms, Part I This course covers the essential information that every serious programmer needs to know about algorithms and data structures, with emphasis on applications and scientific performance analysis of Java implementations. Part I covers elementary data structures, sorting, and searching algorithms. Part II focuses on graph- and string-processing algorithms. All the features of this course are available for free. People who are interested in digging deeper into the content may wish to obtain the textbook Algorithms, Fourth Edition (upon which the course is based) or visit the website algs4.cs.princeton.edu for a wealth of additional material. This course does not offer a certificate upon completion. **Difficulty:** Beginner | **Language:** English | **Duration:** 6 weeks of study, 6–10 hours per week. | **Price:** Free **Link:** https://www.coursera.org/learn/algorithms-part1 **Tags:** courses, technology-computer-science, algorithms ### 4. Introduction to Graph Theory We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. In this online course, among other intriguing applications, we will see how GPS systems find shortest routes, how engineers design integrated circuits, how biologists assemble genomes, why a political map can always be colored using a few colors. We will study Ramsey Theory which proves that in a large system, complete disorder is impossible! By the end of the course, we will implement an algorithm which finds an optimal assignment of students to schools. This algorithm, developed by David Gale and Lloyd S. Shapley, was later recognized by the conferral of Nobel Prize in Economics. As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students. **Difficulty:** Beginner | **Language:** English | **Duration:** 5 weeks, 3-5 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/graphs **Tags:** courses, mathematics-statistics, combinatorics --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/trees