Define a Fenwick tree and its applications.

A Fenwick tree, also known as a Binary Indexed Tree (BIT), is a data structure providing efficient methods for cumulative frequency tables.

A Fenwick tree is a type of binary tree that is used in computer science to solve range queries and update problems efficiently. It was first proposed by Peter M. Fenwick in 1994. The tree is an array representation of a binary tree where each node corresponds to an interval of the input data. It is particularly useful in scenarios where we need to frequently update elements and calculate prefix sums.

The Fenwick tree is based on the concept of binary indexing, which is why it is also known as a Binary Indexed Tree. Each node of the Fenwick tree stores the sum of a range of values from the original array. The range covered by each node is determined by the binary representation of its index in the array. The least significant bit (LSB) in the binary representation of the index determines the range of responsibility of that index.

The Fenwick tree has two main operations: update and prefix sum. The update operation adds a value to an element of the array and updates all the relevant nodes in the Fenwick tree. The prefix sum operation calculates the sum of all elements up to a given index in the array. Both these operations can be performed in O(log n) time, which is a significant improvement over the naive O(n) time for an array or O(n log n) time for a balanced binary search tree.

Applications of the Fenwick tree are numerous. It is widely used in computational geometry to count the number of points in a range, in data compression to compute cumulative frequency tables, in competitive programming for range update and range query problems, and in machine learning for cumulative distribution function computation. It is also used in the implementation of certain algorithms, such as the Schensted algorithm for the longest increasing subsequence problem. The Fenwick tree's ability to perform both updates and queries in logarithmic time makes it a powerful tool in these and many other applications.