[C/C++ Data Structure] Full analysis of clue binary trees: from mathematical principles to C++ implementation

Directory title

  • 1. Introduction
    • 1.1 What is a clue binary tree
    • 1.2 Application scenarios of clue binary trees
  • 2. Mathematical perspective
    • 2.1 Definition of clue binary tree
    • 2.2 Comparison between clue binary tree and ordinary binary tree
    • 2.3 Properties of clue binary trees
  • 3. Data Structure Concepts and Terminology
    • 3.1. Threading of Nodes
      • 3.1.1. Predecessor and Successor Nodes
    • 3.2. Types of Threaded Binary Trees (Types of Threaded Binary Trees)
      • 3.2.1. Full Threaded Binary Tree
      • 3.2.2. Half Threaded Binary Tree
  • 4. C++ Implementation (C++ Implementation)
    • 4.1 Definition of Node Structure
    • 4.2 Implementation of Threading Process
    • 4.3 Traversal of Threaded Binary Tree
    • 4.4 Insertion and Deletion Operations
  • 5. Advantages and Disadvantages
    • 5.1 Advantages
      • 5.1.1 Improve space utilization
      • 5.1.2 Improve traversal efficiency
      • 5.1.3 Simplify certain operations
    • 5.2 Disadvantages
      • 5.2.1 increases programming complexity
      • 5.2.2 The modification operation is complicated
      • 5.2.3 Additional space overhead
  • Real-world Use Cases
    • 6.1 E-commerce Websites
      • 6.1.1 Product Categorization
      • 6.1.2 Search Optimization
    • 6.2 Database Query Optimization
      • 6.2.1 Indexing Structure
    • 6.3 Compiler Design
      • 6.3.1 Expression Optimization
    • 6.4 Computer Graphics
      • 6.4.1 Scene Rendering
  • Conclusion

1. Introduction

1.1 What is a clue binary tree

A clued binary tree is a special type of binary tree in which a null left pointer points to the node’s predecessor and a null right pointer points to the node’s successor. This data structure makes binary tree traversal more efficient, especially for in-order traversal. In a normal binary tree, finding the predecessor or successor of a node may require O(n) time complexity, but in a clued binary tree, this operation can be completed in O(1) time complexity.

The main purpose of a clued binary tree is to improve the efficiency of tree traversal, and it does this by storing extra information. In an ordinary binary tree, when the left subtree of a node is empty, we cannot directly know who its predecessor is; similarly, when the right subtree of a node is empty, we cannot directly know who its successor is. Threaded binary trees solve this problem by storing the predecessor and successor information in these null pointers.

1.2 Application scenarios of clue binary trees

Clue binary trees are very useful in scenarios where frequent in-order traversal is required. For example, in a database, we may need to perform sorted access to stored data, and the clue binary tree provides a fast traversal method. In addition, clue binary trees are also often used in compiler design for expression tree traversal.

From the perspective of human thinking and existence, the clue binary tree reflects our pursuit of efficiency and cherishing time. We always want to get information faster and use resources more efficiently. The clue binary tree is a manifestation of this kind of thinking. It sacrifices space for time efficiency, reflecting a trade-off and choice.

As it is said in “The Way of Programmer Cultivation”: “Your work is how…” This sentence reminds us that when designing data structures and algorithms, we should comprehensively consider their impact on time and space, and make reasonable decisions. trade-off.

2. Mathematical perspective

2.1 Definition of clue binary tree

The clue binary tree is developed on the basis of the binary tree. It adds two pointer fields to each node, which are used to point to the predecessor and successor nodes of the node in a certain traversal order. This data structure allows us to perform tree traversal operations more efficiently.

In a threaded binary tree, two additional pointer fields are added to each node, pointing to the predecessor and successor nodes of that node in a specific traversal order. This data structure enhances the efficiency of tree traversal operations.

2.2 Comparison of clue binary trees and ordinary binary trees

The biggest difference between a clued binary tree and an ordinary binary tree is the pointer field of its node. In an ordinary binary tree, the left and right pointer fields of a node either point to its child nodes or are empty. In a clued binary tree, these pointer fields may point to its child nodes, or to its predecessor or successor nodes in a certain traversal order.

The main difference between threaded binary trees and regular binary trees lies in the pointer fields of the nodes. In a regular binary tree, the left and right pointer fields of a node either point to its children or are null. these pointer fields may point to its children or to its predecessor or successor nodes in a specific traversal order.

2.3 Properties of clue binary trees

The properties of clued binary trees make them efficient in traversal operations. Since each node directly points to its predecessor and successor nodes, we can find the predecessor or successor node of any node in O(1) time complexity, which is impossible in ordinary binary trees.

The properties of threaded binary trees make them efficient in traversal operations. Since each node directly points to its predecessor and successor nodes, we can find the predecessor or successor node of any node in O(1) time complexity, which is not possible in regular binary trees.

3. Data Structure Concepts and Terminology

Before discussing Threaded Binary Tree, we need to understand some basic data structure concepts and terminology. These concepts and terms will help us better understand the structure and working principle of clue binary trees.

3.1. Threading of Nodes

Threading is a process that uses null pointers in binary trees to store the predecessor nodes or successor nodes pointing to nodes in a certain traversal order. The purpose of this is to improve the efficiency of tree traversal.

In a binary tree, each node has at most two child nodes, so each node has two pointer fields. In an ordinary binary tree, if a node has no left (or right) child node, its left (or right) pointer field will be empty. The idea of a clued binary tree is to use these empty pointer fields to store pointers to the predecessor node or successor node of the node under in-order traversal.

3.1.1. Predecessor and Successor Nodes

In binary tree traversal, the predecessor node of a node is the node it visited before, and the successor node is the node it visits after. Clue binary trees use empty pointer fields to store information about these nodes.

3.2. Types of Threaded Binary Trees (Types of Threaded Binary Trees)

There are two main types of clued binary trees: full-clue binary trees and half-clue binary trees.

3.2.1. Full Threaded Binary Tree

In a fully clued binary tree, all empty left pointer fields store pointers to the predecessor nodes of the node under in-order traversal, and all empty right pointer fields store pointers to the successor nodes of the node under in-order traversal. pointer.

3.2.2. Half Threaded Binary Tree

In a semi-clue binary tree, only part of the null pointer field is used to store information about predecessor or successor nodes.

Threading allows us to traverse a binary tree more efficiently, especially to find the predecessor or successor of a node without a parent pointer. This is useful in some applications, such as finding the next larger node (successor) of a node in a binary search tree.

In introducing these concepts, we incorporate deep insights into the human mind and existence. We know that the human brain is always looking for the most efficient path when processing information. The clued binary tree is a manifestation of this way of thinking, creating a more efficient data structure by utilizing otherwise wasted space.

As the “Tao Te Ching” says: “Do nothing and do nothing.” When designing data structures, we should learn to utilize every available resource, even if they seem empty or useless.

In the next chapter, we’ll dive into a C++ implementation of a trailed binary tree, showing how to translate these theoretical concepts into actual runnable code.

4. C++ Implementation (C++ Implementation)

In this chapter, we will delve into the implementation of threaded binary trees in C++. We’ll start with the definition of the node structure and then step through the threading process, traversal methods, and insertion and deletion operations.

4.1 Definition of Node Structure

The nodes of the clue binary tree not only contain data and pointers to the left and right child nodes, but also contain two flag bits to indicate whether the left and right pointers point to child nodes or predecessor/successor nodes.

struct ThreadedTreeNode {<!-- -->
    int data;
    ThreadedTreeNode* left;
    ThreadedTreeNode* right;
    bool leftThreaded;
    bool rightThreaded;
};

In this structure, data stores the value of the node, and left and right point to the left and right child nodes respectively. leftThreaded and rightThreaded are Boolean types. When they are true, it means that the corresponding pointer points to the predecessor or successor node, not the child node. .

4.2 Implementation of Threading Process

Threading is the process of converting a binary tree into a threaded binary tree. We can achieve this process through in-order traversal.

void threadBinaryTree(ThreadedTreeNode* root) {<!-- -->
    if (root == nullptr) return;
    static ThreadedTreeNode* prev = nullptr;
    threadBinaryTree(root->left);
    if (prev != nullptr) {<!-- -->
        if (prev->right == nullptr) {<!-- -->
            prev->right = root;
            prev->rightThreaded = true;
        }
        if (root->left == nullptr) {<!-- -->
            root->left = prev;
            root->leftThreaded = true;
        }
    }
    prev = root;
    threadBinaryTree(root->right);
}

In this function, we use in-order traversal to traverse the binary tree and set clues during the traversal process. The prev variable is used to store the previously visited node.

4.3 Traversal of Threaded Binary Tree

The process of traversing a clue binary tree is similar to traversing an ordinary binary tree, but it should be noted that we need to judge whether we need to visit the predecessor or successor node based on the clue.

void inorderTraversal(ThreadedTreeNode* root) {<!-- -->
    ThreadedTreeNode* curr = root;
    while (curr != nullptr) {<!-- -->
        while (curr->leftThreaded == false) {<!-- -->
            curr = curr->left;
        }
        cout << curr->data << " ";
        while (curr->rightThreaded == true) {<!-- -->
            curr = curr->right;
            cout << curr->data << " ";
        }
        curr = curr->right;
    }
}

In this function, we first find the leftmost node and then visit each node in the order of in-order traversal.

4.4 Insertion and Deletion Operations

Insertion and deletion operations are more complex in threaded binary trees because they require updating the thread. Here we only give a simple example of the insertion operation.

void insert(ThreadedTreeNode* & amp; root, int data) {<!-- -->
    ThreadedTreeNode* newNode = new ThreadedTreeNode{<!-- -->data, nullptr, nullptr, false, false};
    if (root == nullptr) {<!-- -->
        root = newNode;
        return;
    }
    ThreadedTreeNode* curr = root;
    ThreadedTreeNode* parent = nullptr;
    while (curr != nullptr) {<!-- -->
        parent = curr;
        if (data < curr->data) {<!-- -->
            if (curr->leftThreaded == false) {<!-- -->
                curr = curr->left;
            } else {<!-- -->
                break;
            }
        } else if (data > curr->data) {<!-- -->
            if (curr->rightThreaded == false) {<!-- -->
                curr = curr->right;
            } else {<!-- -->
                break;
            }
        } else {<!-- -->
            delete newNode; // Data already exists in the tree
            return;
        }
    }
    if (data < parent->data) {<!-- -->
        newNode->left = parent->left;
        newNode->right = parent;
        parent->leftThreaded = false;
        parent->left = newNode;
    } else {<!-- -->
        newNode->right = parent->right;
        newNode->left = parent;
        parent->rightThreaded = false;
        parent->right = newNode;
    }
}

In this function, we first create a new node, then find the appropriate position to insert it, and update the clue.

In this way, we are not only imparting knowledge, but also cultivating readers’ logical thinking and problem-solving abilities in a subtle way. Through the combination of code and text, we make abstract concepts concrete and visible, and make complex problems simple and easy to understand. This is just like what is said in “The Way of Programmer Cultivation”: “The code is written for people to read, and it can be run on the machine.” Our goal is to allow readers to not only understand the working principle of the clue binary tree, but also master its implementation way, and even be able to improve and optimize it if needed.

In the next section, we will explore the advantages and disadvantages of the clued binary tree and how it performs in practical applications.

5. Advantages and Disadvantages

5.1 Advantages

Threaded Binary Tree is a tree structure that stores the predecessor node or successor node pointing to the node in the null pointer field of the binary tree. The design of this data structure provides a series of advantages.

5.1.1 Improve space utilization

In a normal binary tree, the left and right child pointers of leaf nodes are usually empty, which results in a waste of space. The threaded binary tree improves space utilization by utilizing these null pointer fields to store pointers to the predecessor or successor of the node.

5.1.2 Improve traversal efficiency

In a clued binary tree, the predecessors and successors of a node are directly accessible, making traversing the binary tree more efficient. Especially in inorder traversal, the clued binary tree can be traversed in O(n) time complexity without using a stack or recursion.

5.1.3 Simplify certain operations

Clue binary trees simplify certain tree operations, such as finding the predecessor or successor of a node, which may require complex traversal operations in ordinary binary trees.

5.2 Disadvantages

Although the clued binary tree has its advantages, it also has some disadvantages and limitations.

5.2.1 increases programming complexity

The implementation of clued binary trees is more complex than ordinary binary trees and requires more programming work to maintain clue information. This can be a challenge for beginners.

5.2.2 Complex modification operations

When inserting or deleting operations in a threaded binary tree, thread information needs to be updated, which makes these operations more complex and time-consuming than in a normal binary tree.

5.2.3 Additional space overhead

Although the clue binary tree improves space utilization, it also introduces additional space overhead to store clue information. In each node, additional space is needed to store the predecessor and successor pointers, as well as flag bits that identify clues.

Through the analysis of the advantages and disadvantages of the clue binary tree, we can see that although it provides advantages in terms of space utilization and traversal efficiency, it also comes at the expense of increased programming complexity and operation complexity. Therefore, when choosing to use a threaded binary tree, you need to weigh its advantages and disadvantages based on specific application scenarios and requirements.

Real-world Use Cases

6.1 E-commerce Websites

Clue binary trees play an important role in e-commerce websites, especially in product classification and search optimization. For example, when a user searches for a specific product on the website, the clue binary tree can be used to speed up the search process and quickly locate related product categories.

6.1.1 Product Categorization

In e-commerce websites, products are usually organized into categories. A clue binary tree can be used to store the hierarchical structure of these categories, making it very efficient to quickly jump from one category of products to related categories.

6.1.2 Search Optimization

By using clue binary trees, search algorithms can find the products users need faster and improve user experience. This is because clued binary trees provide a way to quickly traverse the tree structure without recursion, thus speeding up the search process.

6.2 Database Query Optimization

In database management systems, threaded binary trees are used to optimize query operations, especially for complex query conditions. They increase the speed of data retrieval, thereby improving overall performance.

6.2.1 Indexing Structure

The clue binary tree can be used as an index structure to help quickly locate specific records in the database. This structure is particularly suitable for databases that need to be updated frequently, because they can quickly update the index without sacrificing retrieval speed.

6.3 Compiler Design (Compiler Design)

In compiler design, clued binary trees are used to optimize the evaluation of expressions, especially for complex mathematical expressions.

6.3.1 Expression Optimization

By converting expressions into clued binary trees, the compiler can calculate results more efficiently, optimize the calculation process, and increase execution speed.

6.4 Computer Graphics

In computer graphics, threaded binary trees are used to optimize the graphics rendering process and improve rendering efficiency.

6.4.1 Scene Rendering

The clue binary tree can be used to optimize the scene rendering process and improve rendering efficiency by quickly traversing objects in the scene.

Conclusion

In our programming learning journey, understanding is an important step for us to move to a higher level. However, mastering new skills and ideas always requires time and persistence. From a psychological point of view, learning is often accompanied by constant trial and error and adjustment, which is like our brain gradually optimizing its “algorithm” for solving problems.

That’s why when we encounter mistakes, we should view them as opportunities to learn and improve, not just as annoyances. By understanding and solving these problems, we can not only fix the current code, but also improve our programming skills and prevent making the same mistakes in future projects.

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