add: ClimbingStairsTopDown and fix climbingStairsBottomUpOptimized function naming

Author: Jer3myYu

java/Dynamic Programming/ClimbingStairsBottomUpOptimized.java

@@ -0,0 +1,18 @@
+public record ClimbingStairsBottomUpOptimized() {
+    public int climbingStairsBottomUpOptimized(int n) {
+        if (n <= 2) {
+            return n;
+        }
+        // Set 'oneStepBefore' and 'twoStepsBefore' as the base cases.
+        int oneStepBefore = 2;
+        int twoStepsBefore = 1;
+        for (int i = 3; i < n + 1; i++) {
+            // Calculate the number of ways to reach the current step.
+            int current = oneStepBefore + twoStepsBefore;
+            // Update the values for the next iteration.
+            twoStepsBefore = oneStepBefore;
+            oneStepBefore = current;
+        }
+        return oneStepBefore;
+    }
+}

java/Dynamic Programming/ClimbingStairsTopDown.java

@@ -0,0 +1,21 @@
+import java.util.HashMap;
+import java.util.Map;
+
+public class ClimbingStairsTopDown {
+    Map<Integer, Integer> memo = new HashMap<>();
+
+    public int climbingStairsTopDown(int n) {
+        // Base cases: With a 1-step staircase, there's only one way to 
+        // climb it. With a 2-step staircase, there are two ways to climb it.
+        if (n <= 2) {
+            return n;
+        }
+        if (memo.containsKey(n)) {
+            return memo.get(n);
+        }
+        // The number of ways to climb to the n-th step is equal to the sum 
+        // of the number  of ways to climb to step n - 1 and to n - 2.
+        memo.put(n, climbingStairsTopDown(n - 1) + climbingStairsTopDown(n - 2));
+        return memo.get(n);
+    }
+}