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Data Structures
Table of Contents
Unit 5 Overview: Trees and Binary Trees
5.1 Tree Terminology and Properties
5.2 Binary Tree Representation and Traversals
5.3 Tree Applications and Implementations

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5.2 Binary Tree Representation and Traversals

Citation:

Binary trees are fundamental data structures in computer science. They consist of nodes connected in a hierarchical manner, with each node having at most two children. This structure allows for efficient storage and retrieval of data.

Binary trees can be represented using linked structures or arrays. Traversing these trees is crucial for many operations. Three main traversal methods - preorder, inorder, and postorder - offer different ways to visit nodes, each with specific applications in problem-solving.

Binary Tree Representation

Representation of binary trees

Linked structure representation

  • Each node contains data and references to its left and right child nodes
  • Nodes dynamically allocated in memory allows for flexible tree structure
  • Easily insert or delete nodes by updating references

Array representation

  • Binary tree stored in an array with specific indexing rules
  • For a node at index $i$, left child at $2i+1$, right child at $2i+2$
  • Parent of node at index $i$ located at $\lfloor(i-1)/2\rfloor$
  • Requires fixed size array and may waste space for incomplete trees (sparse arrays)

Binary Tree Traversals

Comparison of tree traversal methods

Preorder traversal (NLR: Node, Left, Right)

  • Visit root node first, then recursively traverse left and right subtrees
  • Useful for creating a copy of the tree or prefix expression evaluation

Inorder traversal (LNR: Left, Node, Right)

  • Recursively traverse left subtree, visit root, then traverse right subtree
  • Yields nodes in non-decreasing order for binary search trees (BSTs)

Postorder traversal (LRN: Left, Right, Node)

  • Recursively traverse left and right subtrees, then visit root node
  • Useful for deleting nodes or postfix expression evaluation

Algorithms for tree traversal

Recursive algorithms

  • Use recursive function calls to traverse the tree
  • Base case handles empty tree or null node
  • Recursive case processes current node and calls function on left and right subtrees

Iterative algorithms

  • Use stack or queue data structure to track nodes during traversal
  • Preorder uses stack to simulate recursive calls (push right child first, then left)
  • Inorder uses stack to keep track of visited nodes (push all left children, then backtrack)
  • Postorder uses two stacks or modifies preorder traversal (push root to additional stack after processing)

Applications of tree traversals

Searching for specific value in binary search tree

  • Perform inorder traversal to obtain sorted order of elements (ascending)
  • Binary search can locate value efficiently in $O(\log n)$ time

Evaluating arithmetic expressions represented as binary trees

  • Use postorder traversal to evaluate subexpressions and combine results
  • Operators are non-leaf nodes, operands are leaf nodes (constants or variables)

Serializing and deserializing binary trees

  • Use preorder or postorder traversal to convert tree to linear representation (array or string)
  • Reconstruct tree from serialized data using same traversal order and marker for null nodes

Implementing binary tree algorithms

  • Calculate height or depth of binary tree using postorder traversal (max depth of subtrees + 1)
  • Find lowest common ancestor (LCA) of two nodes using preorder traversal and recursion
  • Check if binary tree is balanced (heights of left and right subtrees differ by at most 1) or a valid BST (inorder traversal yields sorted sequence)

Key Terms to Review (16)

Binary tree representation: Binary tree representation is a method of organizing data in a tree structure where each node has at most two children, referred to as the left child and the right child. This structure enables efficient data storage, retrieval, and manipulation through various traversal techniques, which are essential for working with binary trees in computer science.
Traversal algorithms: Traversal algorithms are procedures used to visit and process each node in a data structure, particularly in trees and graphs, ensuring that all elements are accessed in a systematic manner. These algorithms are essential for performing operations such as searching, sorting, and modifying data within a structure. In the context of binary trees, traversal algorithms dictate the order in which nodes are visited, making them crucial for understanding tree structures and their representations.
Recursive structure: A recursive structure is a data structure that is defined in terms of itself, allowing for the representation of complex data in a hierarchical manner. This concept is central to many data structures, particularly binary trees, where each node contains references to its children, thus creating a self-similar structure that can be easily navigated and manipulated using recursive algorithms.
Insertion algorithm: An insertion algorithm is a method used to add a new element into a data structure while maintaining its properties. In the context of binary trees, this algorithm is essential as it ensures that each element is placed in accordance with the binary search tree rules, where all elements to the left of a node are smaller, and those to the right are larger. This process is vital for keeping the tree balanced and efficient for future operations such as searches and deletions.
Edge: An edge is a fundamental component in graph theory, representing a connection or relationship between two vertices (or nodes) in a graph. In the context of various data structures, edges can signify relationships in hierarchical structures like trees, or connections in more complex networks, playing a critical role in understanding and navigating these structures.
Post-order traversal: Post-order traversal is a method of visiting each node in a tree data structure where the nodes are processed in a specific order: first the left subtree, then the right subtree, and finally the root node. This traversal method is particularly useful for tasks such as deleting a tree or evaluating expressions represented in binary trees, as it ensures that children are processed before their parent nodes.
Pre-order traversal: Pre-order traversal is a method of visiting all the nodes in a tree data structure where the current node is processed before its child nodes. This traversal method is essential for various tree operations, such as creating a copy of the tree, generating prefix expressions for expression trees, and performing operations on binary trees and binary search trees (BSTs). Understanding pre-order traversal helps in exploring the structure of trees and analyzing their properties in different contexts.
Node: A node is a fundamental part of data structures that represents a single unit within a larger structure, typically containing data and links to other nodes. In linked structures, nodes are used to hold values and connect to other nodes, allowing for efficient traversal and manipulation. The way nodes are organized and interconnected plays a crucial role in the overall performance and functionality of various data structures.
In-order traversal: In-order traversal is a method of visiting each node in a binary tree where the nodes are accessed in a specific sequence: left subtree, root node, and then the right subtree. This technique allows for the retrieval of the nodes in a non-decreasing order when applied to a binary search tree, making it essential for operations like sorting and searching.
Balanced binary tree: A balanced binary tree is a type of binary tree where the height of the left and right subtrees of any node differ by at most one. This balance ensures that operations such as insertion, deletion, and lookup can be performed efficiently, typically in O(log n) time complexity. Maintaining this balance is crucial for optimizing performance in various applications, especially in scenarios that involve frequent updates or queries.
Depth: Depth refers to the number of edges from the root node to a specific node in a tree structure. This concept is crucial for understanding how data is organized and accessed in trees, especially in binary trees where depth can affect operations such as searching and traversing. The depth of a node helps determine the overall efficiency of algorithms that manipulate trees and impacts properties like balance and height.
Height: Height is a measure of the longest path from the root node to a leaf node in a tree structure, which reflects how balanced or imbalanced the tree is. It plays a crucial role in determining the efficiency of various tree operations, as a shorter height often leads to faster search, insert, and delete operations.
Binary search tree: A binary search tree (BST) is a data structure that maintains sorted data in a way that allows for efficient insertion, deletion, and lookup operations. In a BST, each node has at most two children, with the left child containing values less than its parent and the right child containing values greater than its parent, ensuring that the tree remains organized and can be searched quickly.
Leaf node: A leaf node is a node in a tree data structure that does not have any child nodes, meaning it is at the bottom of the tree. These nodes play a crucial role in representing the end points of paths in trees, holding actual data or values, and are essential for understanding how data is structured and accessed within tree-based algorithms.
Time Complexity: Time complexity refers to the computational complexity that describes the amount of time it takes to run an algorithm as a function of the length of the input. It is a critical concept that helps in comparing the efficiency of different algorithms, guiding choices about which data structures and algorithms to use for optimal performance.
Space Complexity: Space complexity refers to the amount of memory space required by an algorithm to execute as a function of the size of the input data. It includes both the space needed for variables and the space needed for the input itself. Understanding space complexity helps in choosing the right data structures and algorithms, as it directly impacts performance and resource usage.