树状数组（fenwick tree）

　　树状数组又称芬威克树，概念上是树状，实际上是使用数组实现的，表现为一种隐式数据结构，balabala...详情请见：https://en.wikipedia.org/wiki/Fenwick_tree

　　其中`i += (i & -i)`的相当于求 $ 2^{n-1} , n \in 1,2,3... $，还可以写成`i += (i & (i ^ (i - 1)))`。

　　code:

#include <iostream>
#define LSB(i) (i & -i)
const int M = 10001;
int bit_tree[M];
int freq[M];
int getsum(int index)
{
	int sum = 0;
	for (index += 1; index > 0; index -= LSB(index))
		sum += bit_tree[index];
	return sum;
}
void update(int n, int index, int val)
{
	for (index += 1; index <= n; index += LSB(index))
		bit_tree[index] += val;
}
void init(int freq[], int n)
{
	for (int i = 1; i <= n; i++)
		bit_tree[i] = 0;
	for (int i = 0; i < n; i++)
		update(n, i, freq[i]);
}
int main()
{
	int n;
	std::cin >> n;
	for (int i = 0; i < n; i++)
		std::cin >> freq[i];
	init(freq, n);
	for (int i = 0; i < n; i++)
	std::cout << getsum(i) << std::endl;
	for (int i = 1; i <= n; i++)
		std::cout << bit_tree[i] << ' ';
	return 0;
}

　　一个例子：