Fenwick tree
A Fenwick tree or binary indexed tree is a data structure that can efficiently update elements and calculate prefix sums in a table of numbers. This structure was proposed by Boris Ryabko in 1989with a further modification published in 1992.It has subsequently become known under the name Fenwick tree after , who described this structure in his 1994 article. When compared with a flat array of numbers, the Fenwick tree achieves a much better balance between two operations: element update and prefix sum calculation. A flat array of numbers can either store the elements or the prefix sums. In the first case, computing prefix sums requires linear time; in the second case, updating the array elements requires linear time (in both cases, the other operation can be performed in constant time). Fenwick trees allow both operations to be performed in time. This is achieved by representing the numbers as a tree, where the value of each node in the tree is the prefix sum of the array from the index of the parent (inclusive) up to the index of the node (exclusive). The tree must therefore have one more node than the number of elements in the flat array representation. The tree structure allows the operations of element retrieval, element update, prefix sum, and range sum to be performed using only node accesses. (Wikipedia).
