feat: add the section of the introduction to dynamic programming (#571)

* add the section of the introduction to
dynamic programming

* add a code comments.

codes/cpp/chapter_dynamic_programming/climbing_stairs_backtrack.cpp

@@ -0,0 +1,43 @@
+
+/**
+ * File: climbing_stairs_backtrack.cpp
+ * Created Time: 2023-06-30
+ * Author: Krahets (krahets@163.com)
+ */
+
+#include "../utils/common.hpp"
+
+/* 回溯 */
+void backtrack(vector<int> &choices, int state, int n, vector<int> &res) {
+    // 当爬到第 n 阶时，方案数量加 1
+    if (state == n)
+        res[0]++;
+    // 遍历所有选择
+    for (auto &choice : choices) {
+        // 剪枝：不允许越过第 n 阶
+        if (state + choice > n)
+            break;
+        // 尝试：做出选择，更新状态
+        backtrack(choices, state + choice, n, res);
+        // 回退
+    }
+}
+
+/* 爬楼梯：回溯 */
+int climbingStairsBacktrack(int n) {
+    vector<int> choices = {1, 2}; // 可选择向上爬 1 或 2 阶
+    int state = 0;                // 从第 0 阶开始爬
+    vector<int> res = {0};        // 使用 res[0] 记录方案数量
+    backtrack(choices, state, n, res);
+    return res[0];
+}
+
+/* Driver Code */
+int main() {
+    int n = 9;
+
+    int res = climbingStairsBacktrack(n);
+    cout << "爬 " << n << " 阶楼梯共有 " << res << " 种方案" << endl;
+
+    return 0;
+}