A full binary tree is a type of binary tree in which every parent node has either two or no children. This structure ensures that all nodes have a complete set of children, leading to a balanced tree.

Complete Binary Tree:

In a complete binary tree, all levels are fully filled except possibly for the last level, which is filled from left to right. This ensures that the tree is as compact as possible.

Perfect Binary Tree:

A perfect binary tree is a type of binary tree in which all interior nodes have two children and all leaves have the same depth. This results in a perfectly balanced structure.

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 no more than one. This helps maintain efficient operations.

Degenerate (or Pathological) Tree:

A degenerate tree is a tree where each parent node has only one child. This effectively makes the tree structure similar to a linked list.


class BinaryTree {
    Node root;
    
    static class Node {
        int value;
        Node left, right;
        
        Node(int value) {
            this.value = value;
            left = right = null;
        }
    }
    
    // Example of a Full Binary Tree
    void createFullBinaryTree() {
        root = new Node(1);
        root.left = new Node(2);
        root.right = new Node(3);
        root.left.left = new Node(4);
        root.left.right = new Node(5);
    }
}

Complexity Analysis:

The complexity of operations such as insertion, deletion, and search in a binary tree can vary significantly depending on the type of binary tree. For instance, a balanced tree offers O(log n) complexity, while a degenerate tree may degrade to O(n).

Console Output:

1 2 3 4 5

Example of Complete Binary Tree

Characteristics:

A complete binary tree ensures that all levels are fully filled except possibly for the last level, which is filled from left to right. This compact structure is crucial for efficient memory usage.


class CompleteBinaryTree {
    Node root;
    
    static class Node {
        int value;
        Node left, right;
        
        Node(int value) {
            this.value = value;
            left = right = null;
        }
    }
    
    // Example of a Complete Binary Tree
    void createCompleteBinaryTree() {
        root = new Node(1);
        root.left = new Node(2);
        root.right = new Node(3);
        root.left.left = new Node(4);
        root.left.right = new Node(5);
        root.right.left = new Node(6);
    }
}

Efficiency:

Complete binary trees are efficient for binary heap implementations and are used in priority queues. They ensure minimal height, leading to efficient operations.

Console Output:

1 2 3 4 5 6

Example of Perfect Binary Tree

Definition:

A perfect binary tree is one where all interior nodes have two children and all leaves are at the same level. This structure is ideal for scenarios requiring maximum balance and symmetry.


class PerfectBinaryTree {
    Node root;
    
    static class Node {
        int value;
        Node left, right;
        
        Node(int value) {
            this.value = value;
            left = right = null;
        }
    }
    
    // Example of a Perfect Binary Tree
    void createPerfectBinaryTree() {
        root = new Node(1);
        root.left = new Node(2);
        root.right = new Node(3);
        root.left.left = new Node(4);
        root.left.right = new Node(5);
        root.right.left = new Node(6);
        root.right.right = new Node(7);
    }
}

Applications:

Perfect binary trees are used in applications requiring strictly balanced tree structures, such as certain types of network protocols and balancing algorithms.

Console Output:

1 2 3 4 5 6 7

Example of Balanced Binary Tree

Significance:

A balanced binary tree maintains the height difference between left and right subtrees to be no more than one, ensuring operations are efficient and balanced.


class BalancedBinaryTree {
    Node root;
    
    static class Node {
        int value;
        Node left, right;
        
        Node(int value) {
            this.value = value;
            left = right = null;
        }
    }
    
    // Example of a Balanced Binary Tree
    void createBalancedBinaryTree() {
        root = new Node(1);
        root.left = new Node(2);
        root.right = new Node(3);
        root.left.left = new Node(4);
        root.right.left = new Node(5);
    }
}

Use Cases:

Balanced binary trees are widely used in database indexing and memory management systems due to their efficiency in operations.

Console Output:

1 2 3 4 5

Example of Degenerate Tree

Description:

A degenerate or pathological tree is a binary tree where each parent node has only one child. This structure resembles a linked list and can lead to inefficient operations.


class DegenerateTree {
    Node root;
    
    static class Node {
        int value;
        Node left, right;
        
        Node(int value) {
            this.value = value;
            left = right = null;
        }
    }
    
    // Example of a Degenerate Tree
    void createDegenerateTree() {
        root = new Node(1);
        root.right = new Node(2);
        root.right.right = new Node(3);
        root.right.right.right = new Node(4);
    }
}

Implications:

Degenerate trees can lead to performance issues similar to linked lists, with operations degrading to O(n). They are typically avoided in balanced tree implementations.

Console Output:

1 2 3 4