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dsa-solutions/lc-solutions/2000-2099/2005-subtree-removal-game-with-fibonacci-tree.md

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+---
+id: subtree-removal-game-with-fibonacci-tree
+title: Subtree Removal Game with Fibonacci Tree
+sidebar_label: 2005 Subtree Removal Game with Fibonacci Tree
+tags:
+  - Binary Tree
+  - Dynamic Programming
+  - Game Theory
+  - LeetCode
+description: "This is a solution to the Subtree Removal Game with Fibonacci Tree problem on LeetCode."
+sidebar_position: 2005
+---
+
+## Problem Description
+
+A Fibonacci tree is a binary tree created using the order function `order(n)`:
+
+- `order(0)` is the empty tree.
+- `order(1)` is a binary tree with only one node.
+- `order(n)` is a binary tree that consists of a root node with the left subtree as `order(n - 2)` and the right subtree as `order(n - 1)`.
+
+Alice and Bob are playing a game with a Fibonacci tree with Alice starting first. On each turn, a player selects a node and removes that node and its subtree. The player that is forced to delete the root loses.
+
+Given the integer `n`, return `true` if Alice wins the game or `false` if Bob wins, assuming both players play optimally.
+
+A subtree of a binary tree `tree` is a tree that consists of a node in `tree` and all of this node's descendants. The tree `tree` could also be considered as a subtree of itself.
+
+### Examples
+
+**Example 1:**
+
+```
+Input: `n = 3`
+Output: `true`
+Explanation:
+Alice takes the node `1` in the right subtree.
+Bob takes either the `1` in the left subtree or the `2` in the right subtree.
+Alice takes whichever node Bob doesn't take.
+Bob is forced to take the root node `3`, so Bob will lose.
+Return `true` because Alice wins.
+```
+
+**Example 2:**
+```
+Input: `n = 1`
+Output: `false`
+Explanation:
+Alice is forced to take the root node `1`, so Alice will lose.
+Return `false` because Alice loses.
+```
+
+**Example 3:**
+
+```
+Input: `n = 2`
+Output: `true`
+Explanation:
+Alice takes the node `1`.
+Bob is forced to take the root node `2`, so Bob will lose.
+Return `true` because Alice wins.
+```
+
+### Constraints
+
+- `1 <= n <= 100`
+
+### Approach
+
+The solution to this problem relies on game theory. We need to determine whether Alice can force a win by always making optimal moves. 
+
+The Fibonacci tree grows in a specific way, and we can use dynamic programming to determine the winner for any given `n`.
+
+We can observe that:
+- For `n = 1`, Bob wins because Alice is forced to remove the root node.
+- For `n = 2`, Alice wins because she can remove the single node in the right subtree, forcing Bob to remove the root node.
+
+This pattern can be extended. We'll use a dynamic programming array `dp` where `dp[i]` is `true` if Alice wins with a tree of order `i`, and `false` otherwise.
+
+The recursion relies on two main observations:
+1. If `dp[n-1]` is `false` or `dp[n-2]` is `false`, then `dp[n]` should be `true` because Alice can force Bob into a losing position.
+2. Otherwise, `dp[n]` is `false`.
+
+#### C++
+
+```cpp
+class Solution {
+public:
+    bool aliceWins(int n) {
+        // Initialize a dp array where dp[i] will be true if Alice wins with a tree of order i
+        vector<bool> dp(n + 1, false);
+
+        // Base cases
+        if (n >= 1) dp[1] = false; // Bob wins if there's only one node
+        if (n >= 2) dp[2] = true;  // Alice wins if there are two nodes
+
+        // Fill the dp array based on the previous results
+        for (int i = 3; i <= n; ++i) {
+            dp[i] = !dp[i - 1] || !dp[i - 2];
+        }
+
+        // The result for the given n is stored in dp[n]
+        return dp[n];
+    }
+};
+
+// Example usage
+int main() {
+    Solution solution;
+    // Test the function with examples
+    bool result1 = solution.aliceWins(3); // Expected output: true
+    bool result2 = solution.aliceWins(1); // Expected output: false
+    bool result3 = solution.aliceWins(2); // Expected output: true
+
+    // Print the results
+    printf("Result for n = 3: %s\n", result1 ? "true" : "false");
+    printf("Result for n = 1: %s\n", result2 ? "true" : "false");
+    printf("Result for n = 2: %s\n", result3 ? "true" : "false");
+
+    return 0;
+}
+```
+
+#### Python
+
+```python
+def alice_wins(n: int) -> bool:
+    dp = [False] * (n + 1)
+    if n >= 1:
+        dp[1] = False  # Base case: Bob wins if there's only one node
+    if n >= 2:
+        dp[2] = True   # Base case: Alice wins if there are two nodes
+
+    for i in range(3, n + 1):
+        dp[i] = not dp[i - 1] or not dp[i - 2]
+
+    return dp[n]
+
+# Test the function with examples
+print(alice_wins(3))  # Output: true
+print(alice_wins(1))  # Output: false
+print(alice_wins(2))  # Output: true
+
+```
+