A Fenwick Tree, also known as a Binary Indexed Tree, is a data structure that helps efficiently update and query cumulative frequency tables. You can implement it in TypeScript with an array.

A Fenwick Tree, also known as a Binary Indexed Tree (BIT), is a data structure that provides efficient methods for calculating and updating cumulative frequencies or prefix sums in a dataset. The key advantage of a Fenwick Tree over a simple array is that both updates and queries can be performed in O(log n) time, whereas a naive implementation might require O(n) for either operation. In TypeScript, you can implement a Fenwick Tree using a 1D array, where each element stores a cumulative frequency of its index and earlier elements. The Fenwick Tree allows you to compute prefix sums, as well as update values at specific indices, by efficiently propagating changes up and down the tree. This data structure is particularly useful in scenarios involving large datasets where frequent updates and queries on cumulative data are required, such as financial systems (for processing transactions), gaming leaderboards (for score updates), and competitive programming (for solving range query problems). Understanding and implementing Fenwick Trees can significantly improve the performance of algorithms involving range queries and updates.