computer notes - Data Structures - 19
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The document discusses various aspects of balanced binary search trees, including: 1) Const keyword can be used to mark parameters and return values as constant to prevent unintended modification. 2) AVL trees are binary search trees where the heights of left and right subtrees differ by at most 1. 3) For a binary search tree to be balanced, the heights of left and right subtrees should be close to equal to avoid a skewed or degenerate tree structure.
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BSTNode.Java:
/**
* Node class used for implementing the BST.
*
* DO NOT MODIFY THIS FILE!!
*
* @author CS 1332 TAs
* @version 1.0
*/
public class BSTNode> {
private T data;
private BSTNode left;
private BSTNode right;
/**
* Constructs a BSTNode with the given data.
*
* @param data the data stored in the new node
*/
BSTNode(T data) {
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}
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*
* @return the data
*/
T getData() {
return data;
}
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*
* @return the left child
*/
BSTNode getLeft() {
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}
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*
* @return the right child
*/
BSTNode getRight() {
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}
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*
* @param data the new data
*/
void setData(T data) {
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*
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import java.util.NoSuchElementException;
/**
* Your implementation of a BST.
*/
public class BST> {
/*
* Do not add new instance variables or modify existing ones.
*/
private BSTNode root;
private int size;
/*
* Do not add a constructor.
*/
/**
* Adds the data to the tree.
*
* This must be done recursively.
*
* The new data should become a leaf in the tree.
*
* Traverse the tree to find the appropriate location. If the data is
* already in the tree, then nothing should be done (the duplicate
* shouldn't get added, and size should not be incremented).
*
* Should be O(log n) for best and average cases and O(n) for worst case.
*
* @param data The data to add to the tree.
* @throws java.lang.IllegalArgumentException If data is null.
*/
public void add(T data) {
// WRITE YOUR CODE HERE (DO NOT MODIFY METHOD HEADER)!
}
/**
* Removes and returns the data from the tree matching the given parameter.
*
* This must be done recursively.
*
* There are 3 cases to consider:
* 1: The node containing the data is a leaf (no children). In this case,
* simply remove it.
* 2: The node containing the data has one child. In this case, simply
* replace it with its child.
* 3: The node containing the data has 2 children. Use the SUCCESSOR to
* replace the data. You should use recursion to find and remove the
* successor (you will likely need an additional helper method to
* handle this case efficiently).
*
* Do NOT return the same data that was passed in. Return the data that
* was stored in the tree.
*
* Hint: Should you use value equality or reference equality?
*
* Must be O(log n) for best and average cases and O(n) for worst case.
*
* @param data The data to remove.
* @return The data that was removed.
* @throws java.lang.IllegalArgumentException If data is null.
* @throws java.util.NoSuchElementException If the data is not in the tree.
*/
public T remove(T data) {
// WRITE YOUR CODE HERE (DO NOT MODIFY METHOD HEADER)!
}
/**
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*
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1) The definition of a BST and basic operations like insertion that involve finding the correct place in the tree for the operation.
2) Balancing the tree through rotations to maintain a height difference of at most 1 between subtrees, as in AVL trees.
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BSTNode.Java Node class used for implementing the BST. .pdf
BSTNode.Java:
/**
* Node class used for implementing the BST.
*
* DO NOT MODIFY THIS FILE!!
*
* @author CS 1332 TAs
* @version 1.0
*/
public class BSTNode> {
private T data;
private BSTNode left;
private BSTNode right;
/**
* Constructs a BSTNode with the given data.
*
* @param data the data stored in the new node
*/
BSTNode(T data) {
this.data = data;
}
/**
* Gets the data.
*
* @return the data
*/
T getData() {
return data;
}
/**
* Gets the left child.
*
* @return the left child
*/
BSTNode getLeft() {
return left;
}
/**
* Gets the right child.
*
* @return the right child
*/
BSTNode getRight() {
return right;
}
/**
* Sets the data.
*
* @param data the new data
*/
void setData(T data) {
this.data = data;
}
/**
* Sets the left child.
*
* @param left the new left child
*/
void setLeft(BSTNode left) {
this.left = left;
}
/**
* Sets the right child.
*
* @param right the new right child
*/
void setRight(BSTNode right) {
this.right = right;
}
}
BST.Java:
import java.util.NoSuchElementException;
/**
* Your implementation of a BST.
*/
public class BST> {
/*
* Do not add new instance variables or modify existing ones.
*/
private BSTNode root;
private int size;
/*
* Do not add a constructor.
*/
/**
* Adds the data to the tree.
*
* This must be done recursively.
*
* The new data should become a leaf in the tree.
*
* Traverse the tree to find the appropriate location. If the data is
* already in the tree, then nothing should be done (the duplicate
* shouldn't get added, and size should not be incremented).
*
* Should be O(log n) for best and average cases and O(n) for worst case.
*
* @param data The data to add to the tree.
* @throws java.lang.IllegalArgumentException If data is null.
*/
public void add(T data) {
// WRITE YOUR CODE HERE (DO NOT MODIFY METHOD HEADER)!
}
/**
* Removes and returns the data from the tree matching the given parameter.
*
* This must be done recursively.
*
* There are 3 cases to consider:
* 1: The node containing the data is a leaf (no children). In this case,
* simply remove it.
* 2: The node containing the data has one child. In this case, simply
* replace it with its child.
* 3: The node containing the data has 2 children. Use the SUCCESSOR to
* replace the data. You should use recursion to find and remove the
* successor (you will likely need an additional helper method to
* handle this case efficiently).
*
* Do NOT return the same data that was passed in. Return the data that
* was stored in the tree.
*
* Hint: Should you use value equality or reference equality?
*
* Must be O(log n) for best and average cases and O(n) for worst case.
*
* @param data The data to remove.
* @return The data that was removed.
* @throws java.lang.IllegalArgumentException If data is null.
* @throws java.util.NoSuchElementException If the data is not in the tree.
*/
public T remove(T data) {
// WRITE YOUR CODE HERE (DO NOT MODIFY METHOD HEADER)!
}
/**
* Returns the root of the tree.
*
* For grading purposes only. You shouldn't need to use this met.
Trees in Data Structure
This document defines and provides examples of trees and binary trees. It begins by defining trees as hierarchical data structures with nodes and edges. It then provides definitions for terms like path, forest, ordered tree, height, and multiway tree. It specifically defines binary trees as having two children per node. The document gives examples and properties of binary trees, including full, complete, and binary search trees. It also explains linear and linked representations of binary trees and different traversal methods like preorder, postorder and inorder. Finally, it provides examples of insertion and deletion operations in binary search trees.
Sienna 6 bst
This document discusses binary search trees (BSTs) and balanced binary search trees. It covers BST operations like add, get, remove and their time complexities. It also discusses using BSTs for sorting and maintaining balance through rotations. Specific balanced tree techniques covered are AVL trees and their balance condition requiring left and right subtree height differences of at most 1.
Trees unit 3
The document discusses binary tree representation and traversal methods. It provides two main representations of binary trees - array and linked representations. It also describes different types of binary trees like full, complete, left-skewed, and right-skewed trees. The document then explains three common traversal techniques for binary trees - inorder, preorder, and postorder traversals. Algorithms and code snippets are given for each traversal. Finally, applications of binary tree traversals are discussed along with expression trees and converting infix to postfix notation using a stack.
Computernotes datastructures-22-111227202516-phpapp02
The document discusses deletion in AVL trees. It outlines 5 cases to consider when deleting a node from an AVL tree and describes the actions to take in each case, such as changing balance factors, performing single or double rotations, and continuing up the tree. It also provides examples of each case.
Binary Trees
The document discusses binary trees and their implementation and traversal methods. It defines a binary tree as a tree where each node has at most two children. It describes the common traversal orders of binary trees as inorder, preorder and postorder. It also discusses breadth first traversal and storing binary trees using node structures. Expression trees are described as binary trees used to represent mathematical expressions where leaves are operands and internal nodes are operators.
Trees - Data structures in C/Java
This document discusses binary trees as a data structure for storing hierarchical data. It defines key tree terminology like root, parent, child, and leaf nodes. It provides examples of how binary trees can be used to represent expressions and for file compression. It also introduces a BinaryTreeNode inner class to represent individual nodes, with constructors to initialize the data, left, and right references. Code is provided to build sample trees from these nodes and an example recursive size method is given.
Computer notes - AVL Tree
AVL (Adelson-Velskii and Landis) tree. An AVL tree is identical to a BST except height of the left and right subtrees can differ by at most 1.height of an empty tree is defined to be (–1).
Computer notes - Analysis of Union
Union is clearly a constant time operation. Running time of find(i) is proportional to the height of the tree containing node i.If unions are done by weight (size), the depth of any element is never greater than log2n.
Web-IT Support and Consulting - dBase exports
The document describes a multi-step process for converting multiple dBase files to CSV format for easier use in Excel and SQL Server. It involves using VBA macros in Excel to:
1) Define source and destination folders and file settings like delimiters
2) Loop through all files in the source folder, opening and exporting the data to individual CSV files in the destination folder
3) Close the files and move to the next one until all are converted
Running the macro will automatically convert hundreds of dBase files to CSV in one go. The document provides detailed code explanations and instructions.
Web-IT Support and Consulting - bulk dBase (DBF) exports via Microsoft Excel ...
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computer notes - Data Structures - 30
The document discusses different techniques for improving the efficiency of union-find algorithms, including union-by-size, path compression, and union-by-height. Union-by-size works by making the root of the smaller tree the child of the larger tree during a union operation, keeping tree heights small. Path compression further optimizes find operations by updating the parent pointers along the search path. Together these optimizations allow union-find algorithms to run in almost linear time for practical purposes.
computer notes - Data Structures - 22
The document discusses deletion in AVL trees and outlines 5 cases to consider when deleting nodes from an AVL tree. It also discusses expression trees and parse trees, providing examples of an expression tree for a mathematical expression and a parse tree for an SQL query. Other uses of binary trees mentioned include their use in compilers for expression trees, parse trees, and abstract syntax trees.
Coding Component (50)Weve provided you with an implementation .docx
Coding Component (50%)
We've provided you with an implementation of an unbalanced binary search tree. The tree implements an ordered dynamic set over a generic comparable type T. Supported operations include insertion, deletion, min, max, and testing whether a value is in the set (via the exists method). Because it's a set, duplicates are not allowed, and the insert operation will not insert a value if it is already present.
We have implemented the BST operations in a recursive style. For example, inserting a value into a tree recurses down the tree seeking the correct place to add a new leaf. Each recursive call returns the root of the subtree on which it was called, after making any modifications needed to the subtree to perform the insertion. Deletion is implemented similarly.
Your job is to add the functionality needed to keep the tree balanced using the AVL property. In particular, you will need to
· augment the tree to maintain the height of each of its subtrees, as discussed in Studio;
· compute the balance at the root of a subtree (which is the height of the root's left subtree minus that of its right subtree);
· implement the AVL rebalancing operation, along with the supporting rotation operations; and
· call the height maintenance and rebalancing operations at the appropriate times during insertion and deletion.
Code Outline
There are two main source code files you need to consider, both in the avl package:
· TreeNode.java implements a class TreeNode that represents a node of a binary search tree. It holds a value (the key of the node) along with child and parent pointers. It has a height data member that is currently not used for anything. You should not modify this file, but you need to understand its contents.
· AVLTree.java implements an ordered set as a binary search tree made out of TreeNode objects.
The AVLTree class provides an interface that includes element insertion and deletion, as well as an exists() method that tests whether a value is present in the set. It also offers min() and max() methods. These methods all work as given for (unbalanced) BSTs, using the algorithms we discussed in lecture.
To implement the AVL balancing method, you will need to fill in some missing code to maintain the height of each subtree and perform rebalancing. Look for the 'FIXME' tags in AVLTree.java to see which methods you must modify.
Height Maintenance
You'll need to set the height data member each time a new leaf is allocated in the tree. You can then maintain the height as part of insertion or deletion using the incremental updating strategy you worked out in Studio 10, Part C.
The update procedure updateHeight() takes in a node and updates its height using the heights of its two subtrees. It should run in constant time.
You'll need to call updateHeight() wherever it is needed – in insertion, deletion, and perhaps elsewhere.
Rebalancing
You must implement four methods as part of AVL rebalancing:
· getBalance() computes the balance fact.
08 B Trees
The document discusses B-trees, which are tree data structures used to store large data sets that cannot fit into main memory. B-trees allow for fast retrieval of data by balancing search trees across multiple disk blocks. They work by having internal nodes with up to m children, with leaf nodes containing keys in sorted order. This balancing allows B-trees to provide fast access and search times even for very large datasets spanning multiple disk drives.
Computer notes - Expression Tree
Algorithm to convert postfix expression into an expression tree. We already have an expression to convert an infix expression to postfix. Read a symbol from the postfix expression. If symbol is an operand, put it in a one node tree and push it on a stack
BINARY SEARCH TREE
1) Introduction to Trees.
2) Basic terminologies
3) Binary tree
4) Binary tree types
5) Binary tree representation
6) Binary search tree
7) Creation of a binary tree
8) Operations on binary search tree Trees
Chap 7 binary threaded tree
This document discusses different types of balanced binary search trees including AVL trees, B-trees, and B+-trees. It provides examples of how to construct and perform operations on each type of tree. AVL trees balance after each insertion or deletion by performing tree rotations if needed. B-trees allow nodes to have more than two children and remain balanced by splitting full nodes. B+-trees are like B-trees but also allow sequential record traversal through leaf nodes. Threaded binary trees add threads to leaf nodes for efficient in-order traversal.
Building Scalable Producer-Consumer Pools based on Elimination-Diraction Trees
We present the ED-Tree, a distributed pool structure based on a combination of the elimination-tree and diffracting-tree paradigms, allowing high degrees of parallelism with reduced contention
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BSTNode.Java Node class used for implementing the BST. .pdf
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Web-IT Support and Consulting - bulk dBase (DBF) exports via Microsoft Excel ...
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Coding Component (50)Weve provided you with an implementation .docx
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computer notes - Data Structures - 28
The document discusses equivalence relations and the union-find algorithm. It defines what makes a binary relation an equivalence relation by having the properties of reflexivity, symmetry, and transitivity. It gives examples like electrical connectivity being an equivalence relation. The union-find algorithm can be used to dynamically determine if elements are in the same equivalence class based on the given relations, by performing find and union operations in time proportional to m+n for m finds and n-1 unions.
computer notes - Data Structures - 35
The document discusses various data structures including skip lists, AVL trees, and hashing. It explains that skip lists allow for logarithmic-time operations and are simple to implement. Hashing provides constant-time operations by mapping keys to array indices via a hash function, but collisions must be handled. Common hash functions discussed include summing character codes or converting to a number in a prime base.
computer notes - Data Structures - 36
The document discusses different methods for handling collisions in hash tables, which occur when two keys hash to the same slot. It describes linear probing, where the next empty slot is used to store the colliding key; quadratic probing which uses a quadratic function to determine subsequent slots; and chaining, where each slot contains a linked list of colliding keys. It notes the advantages and disadvantages of each approach.
computer notes - Data Structures - 14
The document discusses recursion and different traversal methods for binary trees. It explains how the call stack is used during recursive function calls and shows examples of preorder and inorder recursive tree traversals. It then describes how to perform non-recursive inorder traversal using an explicit stack. Finally, it introduces level-order traversal, which visits all nodes at each level from left to right before proceeding to the next level.
computer notes - Data Structures - 29
The document discusses using trees and parent pointers to represent sets and solve the dynamic equivalence problem. Each element begins in its own set, represented as a tree with that element as the root. The union operation merges two trees by making one root point to the other. Find returns the root of the tree an element belongs to by traversing parent pointers up the tree. This can be optimized using a technique called union by rank to keep tree heights small.
computer notes - Data Structures - 38
The document discusses three elementary sorting algorithms: selection sort, insertion sort, and bubble sort. It explains their basic mechanisms, provides examples, and analyzes their time complexities. Selection sort and bubble sort have a worst-case time complexity of O(n2) because they require iterating through the entire array multiple times. Insertion sort's worst-case is also O(n2) as it may require shifting over elements each iteration. The document then introduces the idea of divide and conquer algorithms like merge sort which can achieve optimal O(n log n) time complexity.
computer notes - Data Structures - 25
The document describes the build heap algorithm for creating a min heap data structure from an unordered array of keys. The algorithm iterates from the last parent node to the first, calling a percolateDown function to shift nodes and maintain the min heap property as it builds the heap structure from the initial array. It also provides code snippets in C++ for implementing common heap operations like insertion, deletion, and accessing the minimum element.
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computer notes - Data Structures - 19
- 1. Class No.19 Data Structures http://ecomputernotes.com
- 5. Degenerate Binary Search Tree BST for 14, 15, 4, 9, 7, 18, 3, 5, 16, 20, 17 14 15 4 9 7 18 3 5 16 20 17 http://ecomputernotes.com
- 6. Degenerate Binary Search Tree BST for 3 4 5 7 9 14 15 16 17 18 20 14 15 4 9 7 18 3 5 16 20 17 http://ecomputernotes.com
- 7. Degenerate Binary Search Tree BST for 3 4 5 7 9 14 15 16 17 18 20 14 15 4 9 7 18 3 5 16 20 17 Linked List! http://ecomputernotes.com
- 9. Balanced BST Does not force the tree to be shallow . 14 15 18 16 20 17 4 9 7 3 5 http://ecomputernotes.com
- 15. Balanced Binary Tree -1 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 http://ecomputernotes.com
Editor's Notes
- Start of lecture 19
- Start of 20 (Nov 29)
- End of 19