7.3 Trees and Their Properties

Trees are the backbone of many data structures and algorithms. They're like family trees, but for computer science! In this section, we'll explore their unique properties and why they're so useful.

We'll dive into different types of trees, like rooted and binary trees. We'll also learn about cool algorithms for traversing trees and how they're used to solve real-world problems. Get ready to branch out!

Trees: Definition and Properties

Basic Tree Properties

Top images from around the web for Basic Tree Properties

• 

  Binary Search Tree - Basics Behind View original

  Is this image relevant?

• 

  Binary search trees explained · YourBasic View original

  Is this image relevant?

• 

  Binary Search Tree - Basics Behind View original

  Is this image relevant?

• 

  Binary Search Tree - Basics Behind View original

  Is this image relevant?

• 

  Binary search trees explained · YourBasic View original

  Is this image relevant?

1 of 3

Top images from around the web for Basic Tree Properties

• 

  Binary Search Tree - Basics Behind View original

  Is this image relevant?

• 

  Binary search trees explained · YourBasic View original

  Is this image relevant?

• 

  Binary Search Tree - Basics Behind View original

  Is this image relevant?

• 

  Binary Search Tree - Basics Behind View original

  Is this image relevant?

• 

  Binary search trees explained · YourBasic View original

  Is this image relevant?

1 of 3

• A tree is a connected undirected graph with no cycles, consisting of vertices (nodes) and edges connecting pairs of vertices
• The acyclicity property of trees ensures that there is exactly one path between any two vertices, with no loops or cycles (e.g., a family tree or a binary search tree)
• Trees are minimally connected graphs, meaning that removing any edge disconnects the graph into two separate components (subtrees)
• The number of edges in a tree with n vertices is always n-1, which is the minimum number of edges required for connectivity

Vertex Degrees in Trees

• In a tree, the degree of a vertex (the number of edges incident to it) is at most the number of its neighbors
• There is at least one vertex with degree 1 (a leaf) in a tree
• The maximum degree of a vertex in a binary tree is 3 (a root with two children or an internal node with a parent and two children)
• In a k-ary tree, where each vertex has at most k children, the maximum degree of a vertex is k+1

Tree Types and Characteristics

Rooted Trees

• A rooted tree is a tree in which one vertex is designated as the root, establishing a hierarchical structure and parent-child relationships between vertices
• In a rooted tree, each vertex (except the root) has exactly one parent, and the vertices can be organized into levels based on their distance from the root
• The root is at level 0, its children are at level 1, their children are at level 2, and so on
• Rooted trees are used to represent hierarchical structures (e.g., family trees, organizational charts, or directory structures in file systems)

Binary Trees

• A binary tree is a rooted tree in which each vertex has at most two children, referred to as the left child and the right child
• In a binary tree, the subtree rooted at the left child of a vertex is called the left subtree, and the subtree rooted at the right child is called the right subtree
• Binary trees are commonly used in data structures and algorithms (e.g., binary search trees, expression trees, or Huffman coding trees)
• The maximum number of vertices at level $i$ in a binary tree is $2^i$, and the maximum total number of vertices in a binary tree of height $h$ is $2^{h+1}-1$

Spanning Trees

• A spanning tree of a connected graph is a subgraph that includes all the vertices of the original graph and is a tree (i.e., it has no cycles)
• In a weighted graph, a minimum spanning tree is a spanning tree with the minimum total edge weight among all possible spanning trees
• Spanning trees are used in various applications (e.g., network design, clustering algorithms, or optimization problems)
• Kruskal's and Prim's algorithms are commonly used to find minimum spanning trees in weighted graphs

Tree Traversal Algorithms

Depth-First Search (DFS)

• Depth-first search (DFS) explores a tree by traversing as far as possible along each branch before backtracking, typically using a stack data structure
• DFS can be performed in three main orders: pre-order (root, left, right), in-order (left, root, right), and post-order (left, right, root), each providing a different sequence of vertex visits
• Pre-order traversal is used to create a copy of the tree or to get prefix expressions, in-order traversal is used to get infix expressions or to perform binary search tree operations, and post-order traversal is used to delete the tree or to get postfix expressions
• The time complexity of DFS is $O(V+E)$, where $V$ is the number of vertices and $E$ is the number of edges in the tree

Breadth-First Search (BFS)

• Breadth-first search (BFS) explores a tree level by level, visiting all the vertices at a given level before moving to the next level, typically using a queue data structure
• BFS starts at the root and visits all its children, then moves to the next level and visits all the children of the vertices at that level, and so on, until all vertices are visited
• BFS is used to find the shortest path between two vertices in an unweighted graph or to traverse a tree level by level
• The time complexity of BFS is $O(V+E)$, where $V$ is the number of vertices and $E$ is the number of edges in the tree

Trees vs Other Graph Structures

Trees as Special Cases of Graphs

• Trees are a special case of connected acyclic graphs, satisfying specific properties such as having exactly one path between any two vertices
• A forest is a disjoint union of trees, where each connected component of the graph is a tree
• Trees have a simpler structure compared to general graphs, which can have cycles and multiple paths between vertices
• Many graph algorithms (e.g., shortest path algorithms or network flow algorithms) can be simplified or optimized when applied to trees

Applications of Trees

• Trees can be used to represent hierarchical relationships, such as in family trees, organizational charts, or directory structures in file systems
• Spanning trees are subgraphs of connected graphs that include all the vertices and have no cycles, providing a minimally connected structure
• Minimum spanning trees are used in various applications, such as network design, clustering algorithms, and optimization problems
• Trees can be transformed into rooted trees by designating a vertex as the root, establishing parent-child relationships and a hierarchical structure