> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/functions/recursion-1-10.md).

# 递归输出 1-10

### 递归三要素

1. **终止条件**（边界）：什么时候停止
2. **递归调用**：把问题规模变小
3. **返回/处理**：由子问题结果得到当前结果

### 递归输出 1\~n

```cpp
void print(int n) {
    if (n == 0) return;   // 终止条件
    print(n - 1);         // 先处理小的
    cout << n << " ";     // 再输出（正序）
}

void printReverse(int n) {
    if (n == 0) return;
    cout << n << " ";     // 先输出（倒序）
    printReverse(n - 1);
}
```

`print(10)` 输出 `1 2 3 ... 10`；`printReverse(10)` 输出 `10 9 ... 1`。

### 递归与递推

|    | 递归       | 递推    |
| -- | -------- | ----- |
| 方向 | 由大化小     | 由小推大  |
| 实现 | 函数自调用    | 循环    |
| 风险 | 栈溢出、重复计算 | 无明显风险 |

### 递归求阶乘 / 斐波那契

```cpp
long long fact(int n) { return n <= 1 ? 1 : n * fact(n - 1); }

long long fib(int n) {
    if (n <= 2) return 1;
    return fib(n - 1) + fib(n - 2);   // 直接递归会重复计算，n 大时很慢
}
```

### 优化

* 用**记忆化**保存已算过的结果，避免重复。
* 能改递推的尽量改递推，效率更高。


---

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