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Show the process of inserting 9, 1, 2, 8, 10, 5, 4, 3, 7, 6 into an initially empty AVL tree. Then the root is deleted and replaced by the smallest key in the right subtree. Show the tree after each insertion, deletion, and rotation. Indicate the type of each rotation.

Show the process of inserting 9, 1, 2, 8, 10, 5, 4, 3, 7, 6 into an initially empty AVL tree. Then the root is deleted and replaced by the smallest key in the right subtree. Show the tree after each insertion, deletion, and rotation. Indicate the type of each rotation.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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**Transcription for Educational Website:** 4. The depth of a node is the length of the unique path from the root to the node. Consider a full binary search tree with height \( h \) and \( N = 2^{h+1} - 1 \) nodes. What is the average depth of the \( N \) nodes? 5. Show the process of inserting 9, 1, 2, 8, 10, 5, 4, 3, 7, 6 into an initially empty AVL tree. Then the root is deleted and replaced by the smallest key in the right subtree. Show the tree after each insertion, deletion, and rotation. Indicate the type of each rotation.
Transcribed Image Text:**Transcription for Educational Website:** 4. The depth of a node is the length of the unique path from the root to the node. Consider a full binary search tree with height \( h \) and \( N = 2^{h+1} - 1 \) nodes. What is the average depth of the \( N \) nodes? 5. Show the process of inserting 9, 1, 2, 8, 10, 5, 4, 3, 7, 6 into an initially empty AVL tree. Then the root is deleted and replaced by the smallest key in the right subtree. Show the tree after each insertion, deletion, and rotation. Indicate the type of each rotation.
### Understanding Binary Trees and AVL Trees **3. Binary Trees** Binary trees are structured as follows: ```java public class BTNode<E> { private E data; private BTNode<E> left, right; } ``` In this context, a binary tree `t1` is considered a prefix of another binary tree `t2` if `t2` can be transformed into `t1` by removing zero or more leaves from `t2`. To implement this, write a method: ```java public static boolean prefix(BTNode<Integer> t1, BTNode<Integer> t2) ``` This method should determine if `t1` is indeed a prefix of `t2`. **4. Node Depth in Binary Search Trees** The depth of a node is defined as the length of the unique path from the root to that specific node. For a full binary search tree with a height `h` and `N = 2^(h+1) - 1` nodes, you are tasked with finding the average depth of all `N` nodes. **5. AVL Tree Insertions and Deletions** Illustrate the process of inserting elements `9, 1, 2, 8, 10, 5, 4, 3, 7, 6` into an initially empty AVL tree. Following the insertions, if the root is deleted, it should be replaced by the smallest key in the right subtree. Visualize the AVL tree after each insertion, deletion, and rotation with an indication of the type of each rotation.
Transcribed Image Text:### Understanding Binary Trees and AVL Trees **3. Binary Trees** Binary trees are structured as follows: ```java public class BTNode<E> { private E data; private BTNode<E> left, right; } ``` In this context, a binary tree `t1` is considered a prefix of another binary tree `t2` if `t2` can be transformed into `t1` by removing zero or more leaves from `t2`. To implement this, write a method: ```java public static boolean prefix(BTNode<Integer> t1, BTNode<Integer> t2) ``` This method should determine if `t1` is indeed a prefix of `t2`. **4. Node Depth in Binary Search Trees** The depth of a node is defined as the length of the unique path from the root to that specific node. For a full binary search tree with a height `h` and `N = 2^(h+1) - 1` nodes, you are tasked with finding the average depth of all `N` nodes. **5. AVL Tree Insertions and Deletions** Illustrate the process of inserting elements `9, 1, 2, 8, 10, 5, 4, 3, 7, 6` into an initially empty AVL tree. Following the insertions, if the root is deleted, it should be replaced by the smallest key in the right subtree. Visualize the AVL tree after each insertion, deletion, and rotation with an indication of the type of each rotation.
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