Data Structures: Binary Trees

• Data structures are methods of organizing and storing data in a computer so that it can be used efficiently
• Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks
• Data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services
• Common data structures include arrays, linked lists, trees, graphs, and hash tables
• The choice of data structure depends on the specific needs of the application, including factors such as the size of the data, the frequency of access, and the types of operations that will be performed

Binary Trees

• A binary tree is a tree data structure in which each node has at most two children, which are referred to as the left child and the right child
• Binary trees are used to implement binary search trees and binary heaps
• A binary tree is a rooted tree where each node has at most two children
• A full binary tree is a tree in which every node other than the leaves has two children
• A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes in the last level are as far left as possible
• Binary trees can be traversed using depth-first or breadth-first search
• Depth-first search includes pre-order, in-order, and post-order traversals
• Breadth-first search traverses the tree level by level
• Binary search trees are a special type of binary tree where the value of each node is greater than or equal to the value of all nodes in its left subtree and less than or equal to the value of all nodes in its right subtree
• Binary search trees allow for efficient searching, insertion, and deletion of nodes
• Binary heaps are a type of binary tree that satisfies the heap property: the value of each node is greater than or equal to (in a max-heap) or less than or equal to (in a min-heap) the value of its children
• Binary heaps are used to implement priority queues

Graph Algorithms

• Graph algorithms are a set of instructions used to solve problems on graphs
• Graphs are data structures consisting of nodes (vertices) and connections between nodes (edges)
• Common problems include finding the shortest path between two nodes, determining connectivity, and detecting cycles
• Graph algorithms are used in a wide variety of applications, including social networks, transportation networks, and computer networks
• There are several fundamental graph algorithms:
• Breadth-First Search (BFS) explores all the neighbor nodes at the present depth prior to moving on to the nodes at the next depth level
• Depth-First Search (DFS) explores as far as possible along each branch before backtracking
• Dijkstra's Algorithm is used to find the shortest path from a single source node to all other nodes in a graph with non-negative edge weights
• Bellman-Ford Algorithm is used to find the shortest path from a single source node to all other nodes in a graph, even if the graph contains edges with negative weights
• Prim's Algorithm is a greedy algorithm used to find the minimum spanning tree for a weighted undirected graph
• Kruskal's Algorithm is another greedy algorithm to find the minimum spanning tree. It sorts edges by weight and adds them to the tree if they don't form a cycle
• Topological Sort is used to find a linear ordering of nodes in a directed acyclic graph (DAG) such that for every directed edge from node A to node B, node A comes before node B in the ordering

Hash Tables

• Hash tables are data structures that implement an associative array abstract data type, a structure that can map keys to values
• A hash table uses a hash function to compute an index, also called a hash code, into an array of buckets or slots, from which the desired value can be found
• During lookup, the key is hashed, and the resulting hash code is used to index into the table to find the corresponding value
• Hash functions should be designed to distribute keys evenly across the table to minimize collisions
• Collision resolution techniques include separate chaining (each bucket stores a linked list of entries) and open addressing (probing for an empty slot in the table)
• Hash tables provide average-case constant time complexity for insertion, deletion, and lookup operations, assuming a good hash function and a reasonable load factor (ratio of the number of entries to the number of buckets)
• Hash tables are used in many applications, including database indexing, caching, and symbol tables in compilers

Array Implementation

• Arrays are a basic data structure that store a collection of elements of the same data type in contiguous memory locations
• Elements in an array are accessed using their index, which is an integer representing the position of the element in the array
• Arrays can be one-dimensional (a single row of elements) or multi-dimensional (elements arranged in rows and columns)
• Arrays provide constant-time access to elements given their index, but inserting or deleting elements in the middle of an array can be inefficient because it requires shifting other elements
• Arrays have a fixed size, which must be specified when the array is created
• Dynamic arrays, such as ArrayList in Java or vector in C++, automatically resize themselves when the number of elements exceeds the allocated capacity
• Arrays are used in a wide variety of applications, including storing lists of data, implementing matrices, and representing images

Linked Lists

• Linked lists are linear data structures in which elements are stored in nodes, and each node contains a value and a pointer (or link) to the next node in the sequence
• Unlike arrays, linked lists do not store elements in contiguous memory locations
• Linked lists can be singly linked (each node points to the next node) or doubly linked (each node points to both the next and previous nodes)
• Linked lists allow for efficient insertion and deletion of elements at any position, but accessing an element at a specific position requires traversing the list from the beginning
• Common types of linked lists include:
• Singly Linked List: Each node contains data and a pointer to the next node
• Doubly Linked List: Each node contains data and pointers to both the next and previous nodes
• Circular Linked List: The last node points back to the first node
• Linked lists are used in a variety of applications, including implementing stacks, queues, and hash tables

Tree

• A tree is a hierarchical data structure consisting of nodes connected by edges
• The topmost node in a tree is called the root, and each node can have zero or more children
• Nodes with no children are called leaves
• The depth of a node is the number of edges from the root to the node
• The height of a tree is the maximum depth of any node in the tree
• Trees can be used to represent hierarchical relationships, such as file systems, organizational charts, and decision trees
• Common types of trees include:
• Binary Tree: Each node has at most two children
• Binary Search Tree (BST): A binary tree where the value of each node is greater than or equal to the value of all nodes in its left subtree and less than or equal to the value of all nodes in its right subtree
• AVL Tree: A self-balancing binary search tree
• Red-Black Tree: Another type of self-balancing binary search tree
• B-Tree: A self-balancing tree data structure that is optimized for disk-based storage
• Trees can be traversed using depth-first or breadth-first search algorithms