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12.10 Trees

3 min read•june 18, 2024

graphs are a fundamental concept in mathematics and computer science. They're simple yet powerful structures that model hierarchical relationships and connections without cycles. Understanding trees is crucial for solving problems in various fields.

Tree graphs have unique properties that set them apart from other graph types. With one less than vertices and a single between any two nodes, trees provide efficient ways to organize and traverse data. and minimum spanning trees are essential applications in network design and optimization.

Tree Graphs

Key characteristics of trees

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  • Connected graphs with a path between every pair of vertices
  • graphs that do not contain any cycles (closed paths)
  • Number of edges (e)(e) is always one less than the number of vertices (n)(n): e=n−1e = n - 1
    • A tree with 5 vertices will always have 4 edges
  • Unique path exists between any two vertices (no alternate routes)
  • Removing any disconnects the graph into two separate components
  • Adding any edge creates a (destroys the tree property)
  • Trees have a hierarchical structure with a single node at the top

Methods for spanning tree construction

  • is a that includes all vertices of the original graph
    • Spans all vertices while maintaining a tree structure (no cycles)
  • Path-finding method:
    1. Choose any starting
    2. Traverse the graph (depth-first or )
    3. Add each visited edge to the spanning tree
    4. Stop when all vertices have been visited
  • Edge removal method:
    1. Start with the original graph
    2. Repeatedly remove edges that are part of cycles
    3. Stop when no more cycles remain
    • The remaining graph is a spanning tree

Kruskal's algorithm for minimum spanning trees

  • Finds the () of a
    • MST is the spanning tree with the minimum total edge weight
  • Algorithm steps:
    1. Sort all edges by non-decreasing weight
    2. Create a FF (set of trees), each vertex as a separate tree
    3. Create an empty set SS to store MST edges
    4. While SS has less than n−1n-1 edges and FF has multiple components:
      • Remove the smallest weight edge ee from the sorted list
      • If adding ee to FF does not create a :
        • Add ee to SS
        • Merge the trees in FF connected by ee
    5. Return SS, the set of edges forming the minimum spanning tree
  • Greedily selects the smallest weight edges that maintain a tree structure
    • Guarantees the minimum total edge weight among all spanning trees

Tree Structure and Terminology

  • Root: The topmost node in a tree, from which all other nodes descend
  • : A node with no children, typically found at the bottom of the tree
  • : An edge connecting two nodes in the tree
  • : The number of children a node has
  • : The length of the longest path from the root to a leaf
  • : A tree in which each node has at most two children
  • : The process of visiting all nodes in a tree in a specific order (e.g., pre-order, in-order, post-order)

Key Terms to Review (30)

Acyclic: Acyclic refers to a graph that contains no cycles. In other words, it is a graph where there is no way to start at one vertex and return to it by following the edges.
Acyclic Graph: An acyclic graph is a type of graph that does not contain any cycles, meaning there is no path that starts and ends at the same vertex. This characteristic makes acyclic graphs particularly useful in various applications, including representing hierarchical structures and relationships. One of the most common forms of an acyclic graph is a tree, which is a special type of acyclic graph that has a single connected component and no cycles.
Binary tree: A binary tree is a data structure in which each node has at most two children, referred to as the left child and the right child. This simple yet powerful structure enables efficient data organization and retrieval, making it a fundamental concept in computer science. Binary trees can be used for various applications such as expression parsing, searching algorithms, and representing hierarchical data.
Branch: In the context of tree diagrams and trees, a branch is a line or segment that connects nodes and represents the relationship between them. Each branch signifies a possible outcome or decision point, illustrating how different choices can lead to various results. The structure of branches helps organize information clearly, making it easier to analyze complex scenarios.
Breadth-first search: Breadth-first search is an algorithm used to traverse or search through tree or graph data structures, exploring all the neighbor nodes at the present depth prior to moving on to nodes at the next depth level. This method is effective in finding the shortest path in unweighted graphs and ensures that nodes are visited level by level, which makes it particularly useful in various applications such as networking and puzzle solving.
Connected graph: A connected graph is a type of graph where there is a path between every pair of vertices, meaning all vertices are reachable from one another. This property is essential in understanding various concepts in graph theory, particularly in analyzing routes, networks, and the structure of trees. Connected graphs can have implications for finding Euler circuits and trails, as well as play a critical role in understanding the structure and properties of trees.
Cycle: A cycle in graph theory is a path that starts and ends at the same vertex with no other vertices repeated. It forms a closed loop or circuit within the graph structure.
Cycle: A cycle in graph theory refers to a path that starts and ends at the same vertex, containing at least one edge, with no other repetitions of vertices and edges. This concept is crucial for understanding how graphs are structured and navigated, as cycles can affect various properties and operations within a graph, including traversal and connectivity.
Degree: In mathematics, a degree is a unit of measurement for angles, denoting the size of the angle in a circle. It also represents the number of edges connected to a vertex in graph theory, providing insight into the structure and navigation of graphs. Understanding degrees helps clarify relationships between angles and graph structures, which is essential for analyzing various mathematical concepts.
Depth-first search: Depth-first search (DFS) is an algorithm used to traverse or search through graph structures by exploring as far down a branch as possible before backtracking. This approach allows for the exploration of all possible paths, making it particularly useful in finding Hamilton paths and navigating trees, where each node represents a potential solution or a step in the exploration process.
Directed path: A directed path in a graph is a sequence of vertices where each adjacent pair is connected by a directed edge, following the direction from one vertex to the next. The order of vertices and the direction of edges are crucial for defining such a path.
Edge: An edge is a connection between two vertices in a graph. It can be directed (with a direction) or undirected (without a direction).
Edge: An edge is a fundamental component of a graph that represents a connection or relationship between two vertices (or nodes). Edges can be directed or undirected, and they can have weights associated with them, representing the cost or distance between the connected vertices. Understanding edges is crucial for analyzing graph structures, navigating graphs, and exploring various properties like circuits and paths.
Forest: A forest is a disjoint union of trees, meaning it is a graph without cycles where each connected component is a tree. Forests are acyclic and can consist of one or more isolated trees.
Forest: In graph theory, a forest is a disjoint union of trees, which are acyclic connected graphs. This means that a forest consists of multiple trees, and each tree is a connected component without cycles. The concept of a forest plays a crucial role in understanding the properties of trees and their applications in various fields, such as computer science and network design.
Height: Height refers to the measurement of an object from its base to its topmost point. This dimension is crucial in various geometric contexts, influencing calculations related to area, volume, and surface area. Understanding height is essential for determining the size and capacity of three-dimensional shapes, as well as for analyzing hierarchical structures in data representation.
Kruskal’s algorithm: Kruskal’s algorithm is a method used to find the minimum spanning tree of a connected, undirected graph. It adds edges in increasing order of weight without forming any cycles until all vertices are connected.
Kruskal's algorithm: Kruskal's algorithm is a method used in graph theory to find the minimum spanning tree for a connected, undirected graph. This algorithm operates by sorting all the edges of the graph in non-decreasing order of their weights and adding them one by one to the spanning tree, ensuring that no cycles are formed. It is particularly effective for sparse graphs and plays a crucial role in network design and optimization problems.
Leaf: In the context of trees, a leaf is a terminal vertex that has no children, representing the endpoints of the tree structure. Leaves play an essential role in the overall functionality and efficiency of trees, often holding important data or representing the final output in various applications such as algorithms and data organization.
Minimum spanning tree: A minimum spanning tree (MST) is a subset of edges in a weighted undirected graph that connects all the vertices together without any cycles and with the minimum possible total edge weight. This concept is crucial for optimizing network designs, such as minimizing costs in telecommunications or transportation systems while ensuring all points are connected.
MST: An MST, or Minimum Spanning Tree, is a subset of edges in a connected, undirected graph that connects all vertices together without any cycles and with the minimum possible total edge weight. MSTs are essential in various applications such as network design, clustering, and minimizing costs for connecting points. They provide a way to maintain connectivity while ensuring efficiency in terms of weight or cost associated with the edges.
Path: In the context of graph theory and related fields, a path is a sequence of edges that connects a sequence of vertices without revisiting any vertex. Paths are essential for understanding various structures and algorithms within graph representation, as they reveal how different points are interconnected. They play a crucial role in navigating graphs and analyzing routes in different scenarios.
Root: In the context of trees, a root is the topmost node in a tree structure, serving as the starting point from which all other nodes branch out. It is the ancestor of all nodes in the tree and plays a critical role in defining the structure and organization of the tree. The root connects to child nodes and determines the hierarchy of relationships within the entire tree.
Spanning tree: A spanning tree of a graph is a subgraph that includes all the vertices of the original graph and is connected, without forming any cycles. This means that a spanning tree retains the structure of the original graph while simplifying it to ensure that there are no closed loops, making it essential for various applications like network design and optimization.
Spanning trees: A spanning tree is a subgraph of a connected, undirected graph that includes all the vertices with the minimum possible number of edges. It contains no cycles and connects all the vertices together.
Subgraph: A subgraph is a subset of a graph that includes some of its vertices and the edges connecting them. It retains the structure of the original graph while focusing on a smaller portion of it, which can be useful for analyzing specific parts or properties of the graph. Understanding subgraphs is essential in examining relationships within trees, as they can represent branches or sections of the overall tree structure.
Traversal: Traversal refers to the process of visiting each node in a graph or tree data structure exactly once in a systematic manner. This concept is crucial in understanding how to navigate through complex structures like graphs and trees, enabling various operations such as searching, sorting, and pathfinding.
Tree: A tree is a special type of graph that is connected and acyclic, meaning it has no cycles and there is exactly one path between any two vertices. Trees are fundamental structures in mathematics and computer science, as they can represent hierarchical relationships and are used in various applications, such as data organization and network design. Each tree consists of nodes connected by edges, with one node designated as the root, from which all other nodes branch out.
Vertex: A vertex is a point where two or more curves, lines, or edges meet. In different contexts, it can represent a significant feature such as the peak of a parabola, a corner of a polygon, or a key point in graph theory. Understanding the concept of a vertex helps in analyzing the properties and relationships of various mathematical structures.
Weighted graph: A weighted graph is a type of graph where each edge has an associated numerical value, known as a weight. These weights typically represent costs, distances, or other measures that quantify the relationship between the connected vertices. In this context, weighted graphs are essential for analyzing and optimizing paths and networks, making them applicable in various fields like transportation, computer networking, and logistics.
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