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question:You are given a binary tree and an integer K. Your task is to find the length of the longest path where each node's value is distinct and the sum of the values does not exceed K. # Input Format - The first line contains an integer T, denoting the number of test cases. - Each test case starts with an integer N, the number of nodes in the binary tree. - The next line contains N integers, the values of the nodes. - The next line contains N-1 pairs of integers, each representing an edge between two nodes. # Output Format For each test case, print the length of the longest valid path where each node's value is distinct and the sum of the values does not exceed K. # Constraints - (1 ≤ T ≤ 10) - (1 ≤ N ≤ 1000) - (0 ≤ text{Node Value} ≤ 1000) - (0 ≤ K ≤ 10000) # Sample Input 1 ``` 1 5 10 20 30 40 50 1 2 1 3 2 4 2 5 80 ``` # Sample Output 1 ``` 3 ``` # Explanation 1 Test Case 1: The binary tree is: ``` 10 / 20 30 / 40 50 ``` One possible valid path: 10 → 20 → 30. The sum is 10 + 20 + 30 = 60 which is less than 80 and the length is 3. Another valid path could be 20 → 10 → 30. But, the longest path length remains 3. # Sample Input 2 ``` 2 3 1 2 3 1 2 2 3 2 3 1 1 1 1 2 1 3 1 ``` # Sample Output 2 ``` 1 1 ``` # Explanation 2 Test Case 1: The binary tree is: ``` 1 / 2 / 3 ``` The longest valid path where the sum ≤ 2 is just node 1 itself. Length = 1. Test Case 2: The binary tree is: ``` 1 / 1 1 ``` The only possible paths are single nodes since adding any other node would make the sum > 1. Length = 1.

answer:class TreeNode: def __init__(self, value=0): self.value = value self.left = None self.right = None def build_tree(nodes, edges): if not nodes: return None node_dict = {i + 1: TreeNode(value) for i, value in enumerate(nodes)} for u, v in edges: if node_dict.get(u).left is None: node_dict[u].left = node_dict[v] else: node_dict[u].right = node_dict[v] return node_dict[1] def longest_path_sum_k(root, K): def dfs(node, current_sum, visited): if not node or node.value in visited or current_sum + node.value > K: return 0 visited.add(node.value) current_len = 1 left_len = dfs(node.left, current_sum + node.value, visited) right_len = dfs(node.right, current_sum + node.value, visited) visited.remove(node.value) return current_len + max(left_len, right_len) return dfs(root, 0, set()) def process_input_and_solve(input_str): input_data = input_str.strip().split('n') t = int(input_data[0]) index = 1 results = [] for _ in range(t): n = int(input_data[index]) values = list(map(int, input_data[index + 1].split())) edges = [] for i in range(n - 1): u, v = map(int, input_data[index + 2 + i].split()) edges.append((u, v)) K = int(input_data[index + 1 + n]) index += 2 + n root = build_tree(values, edges) result = longest_path_sum_k(root, K) results.append(result) for res in results: print(res) # Example usage: input_str = 1 5 10 20 30 40 50 1 2 1 3 2 4 2 5 80 process_input_and_solve(input_str)

question:Anna is planning to plant a garden and she wants to arrange the plants in such a way that the total area covered by the plants is maximized. She has a rectangular garden with a length `L` and width `W`, and she can plant either squares of side 1 unit or circles with a diameter of 1 unit. Each square covers an area of 1 square unit, while each circle covers an area of approximately 0.7854 square units. Given the dimensions of the garden, your task is to determine the maximum total area that can be covered by planting either squares or circles. Input: - The first line contains two integers `L` and `W` (1 ≤ L, W ≤ 1000) representing the length and width of the garden, respectively. Output: - Output a single floating-point number representing the maximum area that can be covered with squares or circles, rounded to 4 decimal places. Examples Input: 3 3 Output: 9.0000 Input: 2 4 Output: 8.0000

answer:def max_garden_area(L, W): Calculate the maximum area that can be covered by planting either squares or circles. Parameters: L (int): The length of the garden. W (int): The width of the garden. Returns: float: The maximum area covered, rounded to 4 decimal places. # Total area of the garden total_area = L * W # Area covered with squares max_area_squares = total_area # Area covered with circles circle_area = 3.1416 * (0.5)**2 # using πr^2 where r = 0.5 max_area_circles = (L * W) * circle_area # The maximum area covered is the maximum of the two max_area_covered = max(max_area_squares, max_area_circles) return round(max_area_covered, 4)

question:You are given a list of non-negative integers representing weights [w1, w2, ..., wn]. Determine whether it is possible to partition these weights into two subsets such that the sum of weights in both subsets is equal. Example 1: Input: [1, 5, 11, 5] Output: True Explanation: The array can be partitioned as [1, 5, 5] and [11]. Example 2: Input: [1, 2, 3, 5] Output: False Explanation: The array cannot be partitioned into equal sum subsets. Note: 1. Each of the array element will not exceed 100. 2. The array size will not exceed 200.

answer:def can_partition(nums): Determines if the list of weights can be partitioned into two subsets with equal sum. Args: nums: List[int] : list of non-negative integers representing weights Returns: bool : True if partition exists, False otherwise total_sum = sum(nums) if total_sum % 2 != 0: return False target = total_sum // 2 n = len(nums) dp = [False] * (target + 1) dp[0] = True for num in nums: for i in range(target, num - 1, -1): dp[i] = dp[i] or dp[i - num] return dp[target]

question:String Rotation Check Given two strings, S_1 and S_2, determine if S_2 is a rotation of S_1. Input Two strings S_1 and S_2 separated by a space character are given in a line. Output Print "Yes" if S_2 is a rotation of S_1. Otherwise, print "No". Constraints * 1 leq |S_1|, |S_2| leq 10^5 * Both strings contain only lowercase English letters ('a'-'z'). Sample Input 1 hello lohel Sample Output 1 Yes Sample Input 2 waterbottle erbottlewat Sample Output 2 Yes Sample Input 3 apple pleap Sample Output 3 Yes Sample Input 4 banana ananab Sample Output 4 Yes Sample Input 5 abcd abdc Sample Output 5 No Example Input hello lohel Output Yes

answer:def is_rotation(S1, S2): Determines if S2 is a rotation of S1. Args: S1 (str): The original string. S2 (str): The string to check for rotation. Returns: str: "Yes" if S2 is a rotation of S1, otherwise "No". if len(S1) != len(S2): return "No" combined = S1 + S1 if S2 in combined: return "Yes" else: return "No"