Problem: Easy and hard versions are actually different problems, so we advise you to read both statements carefully.

You are given a weighted rooted tree, vertex 1 is the root of this tree.

A tree is a connected graph without cycles. A rooted tree has a special vertex called the root. A parent of a vertex v is the last different from v vertex on the path from the root to the vertex v. Children of vertex v are all vertices for which v is the parent. A vertex is a leaf if it has no children. The weighted tree is such a tree that each edge of this tree has some weight.

The weight of the path is the sum of edges weights on this path. The weight of the path from the vertex to itself is 0.

You can make a sequence of zero or more moves. On each move, you select an edge and divide its weight by 2 rounding down. More formally, during one move, you choose some edge i and divide its weight by 2 rounding down (w_i := \left⌊(w_i)/(2)\right⌋).

Your task is to find the minimum number of moves required to make the sum of weights of paths from the root to each leaf at most S. In other words, if w(i, j) is the weight of the path from the vertex i to the vertex j, then you have to make ∑_{v ∈ leaves} w(root, v) ≤ S, where leaves is the list of all leaves.

You have to answer t independent test cases.

Input

The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.

The first line of the test case contains two integers n and S (2 ≤ n ≤ 10^5; 1 ≤ S ≤ 10^{16}) — the number of vertices in the tree and the maximum possible sum of weights you have to obtain. The next n-1 lines describe edges of the tree. The edge i is described as three integers v_i, u_i and w_i (1 ≤ v_i, u_i ≤ n; 1 ≤ w_i ≤ 10^6), where v_i and u_i are vertices the edge i connects and w_i is the weight of this edge.

It is guaranteed that the sum of n does not exceed 10^5 (∑ n ≤ 10^5).

Output

For each test case, print the answer: the minimum number of moves required to make the sum of weights of paths from the root to each leaf at most S.

Example

Input


3
3 20
2 1 8
3 1 7
5 50
1 3 100
1 5 10
2 3 123
5 4 55
2 100
1 2 409


Output


0
8
3

Solution (in JAVA):

Solution:
import java.io.*;
import java.math.*;
import java.util.*;

/**
 *
 * @author Saju
 *
 */

public class Main {

    private static int dx[] = { 1, 0, -1, 0 };
    private static int dy[] = { 0, -1, 0, 1 };

    private static final long INF = Long.MAX_VALUE;
    private static final int INT_INF = Integer.MAX_VALUE;
    private static final long NEG_INF = Long.MIN_VALUE;
    private static final int NEG_INT_INF = Integer.MIN_VALUE;
    private static final double EPSILON = 1e-10;

    private static final long MAX = (long) 1e12;
    
    private static final long MOD = 100000007;

    private static final int MAXN = 300005;
    private static final int MAXA = 2000007;
    private static final int MAXLOG = 22;
    private static final double PI = Math.acos(-1);
    
	public static void main(String[] args) throws IOException {

		InputReader in = new InputReader(System.in);
//		Scanner in = new Scanner(System.in);
		PrintWriter out = new PrintWriter(System.out);

//         InputReader in = new InputReader(new FileInputStream("src/test.in"));
//         PrintWriter out = new PrintWriter(new BufferedWriter(new FileWriter("src/test.out")));

		
		
/*


*/		
		
		int test = in.nextInt();
		for (int t = 1; t <= test; t++) {
			int n = in.nextInt();
			long s = in.nextLong();
			
			List<Edge> edges = new ArrayList<>();
			List<Edge> adj[] = new ArrayList[n + 1];
			for(int i = 0; i <= n; i++) {
				adj[i] = new ArrayList<>();
			}
			for(int i = 0; i < n - 1; i++) {
				int u = in.nextInt();
				int v = in.nextInt();
				long w = in.nextLong();
				int c = in.nextInt();
				adj[u].add(new Edge(i, u, v, w));
				adj[v].add(new Edge(i, v, u, w));
				
				Edge e = new Edge(w, c);
				edges.add(e);
			}
			int[]leafs = new int[n + 1];
			dfs(1, 1, adj, edges, leafs);
			
//			int ans = 0;
//			PriorityQueue<Pair> pq = new PriorityQueue<>(Collections.reverseOrder());
//			long sum = 0;
//			int c = 0;
//			for(Edge e : edges) {
//				s -= (e.w * e.cnt);
//				Pair p = new Pair();
//				p.x = (e.w + 1) / 2L * e.cnt;
//				p.y = c++;
//				p.c = e.c;
//				pq.add(p);
////				System.out.println(e.c);
//			}
//			s -= sum;
////			System.out.println(s);
//			while(true) {
//				if(s >= 0) {
//					break;
//				}
//				Pair p = pq.remove();
//				s += p.x;
//				edges.get(p.y).w /= 2L;
//				p.x = (edges.get(p.y).w + 1) / 2L * edges.get(p.y).cnt;
//				pq.add(p);
//				ans += p.c;
//			}
//			out.println(ans);
			
			List<Long> list1 = get(1, s, edges);
			List<Long> list2 = get(2, s, edges);
            int j = list2.size() - 1;
            int ans = INT_INF;;
            for (int i = 0; i < list1.size(); ++i) {
                while (j - 1 >= 0 && list1.get(i) + list2.get(j - 1) <= s) --j;
                if (list1.get(i) + list2.get(j) <= s) {
                    ans = Math.min(ans, i + j * 2);
                }
            }
            out.println(ans);
			
		}
		
		
		
		in.close();
		out.flush();
		out.close();
		System.exit(0);
	}
	
   private static List<Long> get(int cost, long S, List<Edge> edges) {
        List<Long> list = new ArrayList<>();
        PriorityQueue<Pair> pq = new PriorityQueue<>(Collections.reverseOrder());
        long sum = 0;
        for (Edge e : edges) {
           long x = 1L * (e.w + 1) / 2 * e.cnt;
            if (e.c == cost) {
                sum += 1L * e.w * e.cnt;
                Pair p = new Pair();
                p.w = e.w;
                p.x = x; 
                p.y = e.cnt;
                
                pq.add(p);
            }
        }
        list.add(sum);
        while (!pq.isEmpty() && pq.peek().x > 0) {
            Pair top = pq.poll();
            sum -= top.x;
            top.w /= 2;
            top.x = 1L * (top.w + 1) / 2 * top.y;
            pq.add(top);
            list.add(sum);
           
        }
        return list;
    }
	
	private static void dfs(int u, int par, List<Edge>[] adj, List<Edge> edges, int[] leafs) {
		boolean isLeaf = true;
		for(Edge e : adj[u]) {
			int v = e.v;
			if(v == par) {
				continue;
			}
			isLeaf = false;
			dfs(v, u, adj, edges, leafs);
			edges.get(e.index).cnt += leafs[v];
			leafs[u] += leafs[v];
		}
		if(isLeaf) {
			leafs[u]++;
		}
	}
	
	private static class Pair implements Comparable<Pair>{
		long w;
		long x;
		long y;
		Pair(){
			
		}
		
		@Override
		public int compareTo(Pair o) {
			int val = Long.compare(x, o.x);
			return val;
		}
	}

	private static class Edge{
		int u;
		int v;
		long w;
		int index;
		long cnt = 0;
		int c;
		Edge(int index, int u, int v, long w){
			this.index = index;
			this.u = u;
			this.v = v;
			this.w = w;
			this.cnt = 0;
		}
		public Edge(long w, int c) {
			this.w = w;
			this.cnt = 0;
			this.c = c;
		}
		
	}
	
	
	/*
	 * return the number of elements in list that are less than or equal to the val
	 */
	private static long upperBound(List<Long> list, long val) {
		int start = 0;
		int len = list.size();
		int end = len - 1;
		int mid = 0;

		while (true) {
			if (start > end) {
				break;
			}
			mid = (start + end) / 2;
			long v = list.get(mid);
			if (v == val) {
				start = mid;
				while(start < end) {
					mid = (start + end) / 2;
					if(list.get(mid) == val) {
						if(mid + 1 < len && list.get(mid + 1) == val) {
							start = mid + 1;
						}
						else {
							return mid + 1;
						}
					}
					else {
						end = mid - 1;
					}
				}
				return start + 1;
			}
			if (v > val) {
				end = mid - 1;
			} else {
				start = mid + 1;
			}
		}
		if (list.get(mid) < val) {
			return mid + 1;
		}
		return mid;
	}


	private static boolean isPalindrome(String str) {
		StringBuilder sb = new StringBuilder();
		sb.append(str);
		String str1 = sb.reverse().toString();
		return str.equals(str1);
	}

	private static String getBinaryStr(long n, int j) {
		String str = Long.toBinaryString(n);
		int k = str.length();
		for (int i = 1; i <= j - k; i++) {
			str = "0" + str;
		}

		return str;
	}
	
	private static long modInverse(long r) {
		return bigMod(r, MOD - 2, MOD);
	}
	
	private static long bigMod(long n, long k, long m) {

        long ans = 1;
        while (k > 0) {
            if ((k & 1) == 1) {
                ans = (ans * n) % m;
            }
            n = (n * n) % m;
            k >>= 1;
        }
        return ans;
    }
    
	private static long ceil(long n, long x) {
		long div = n / x;
        if(div * x != n) {
            div++;
        }
        return div;
	}
	
    private static int ceil(int n, int x) {
        int div = n / x;
        if(div * x != n) {
            div++;
        }
        return div;
    }

    
    private static int abs(int x) {
        if (x < 0) {
            return -x;
        }
        return x;
    }
    
    private static double abs(double x) {
        if (x < 0) {
            return -x;
        }
        return x;
    }

    private static long abs(long x) {
        if(x < 0) {
            return -x;
        }
        return x;
    }

    private static int lcm(int a, int b) {
        return (a * b) / gcd(a, b);
    }

    private static int gcd(int a, int b) {
        if (a == 0)
            return b;
        return gcd(b % a, a);
    }
    
    private static long gcd(long a, long b) {
        if (a == 0)
            return b;
        return gcd(b % a, a);
    }

    private static int log(long x, int base) {
        return (int) (Math.log(x) / Math.log(base));
    }

    private static long min(long a, long b) {
        if (a < b) {
            return a;
        }
        return b;
    }

    private static int min(int a, int b) {
        if (a < b) {
            return a;
        }
        return b;
    }

    private static long max(long a, long b) {
        if (a < b) {
            return b;
        }
        return a;
    }

    private static int max(int a, int b) {
        if (a < b) {
            return b;
        }
        return a;
    }

    private static class InputReader {
        public BufferedReader reader;
        public StringTokenizer tokenizer;

        public InputReader(InputStream stream) {
            reader = new BufferedReader(new InputStreamReader(stream));
            tokenizer = null;
        }

        public String next() {
            try {
                while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                    tokenizer = new StringTokenizer(reader.readLine());

                }
            } catch (IOException e) {
                return null;
            }
            return tokenizer.nextToken();
        }

        public String nextLine() {
            String line = null;
            try {
                tokenizer = null;
                line = reader.readLine();
            } catch (IOException e) {
                throw new RuntimeException(e);
            }
            return line;
        }

        public int nextInt() {
            return Integer.parseInt(next());
        }

        public double nextDouble() {
            return Double.parseDouble(next());
        }

        public long nextLong() {
            return Long.parseLong(next());
        }

        public boolean hasNext() {
            try {
                while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                    tokenizer = new StringTokenizer(reader.readLine());
                }
            } catch (Exception e) {
                return false;
            }
            return true;
        }
        
        public int[] nextIntArr(int n) {
        	int arr[] = new int[n];
    		for(int i = 0; i < n; i++) {
    			arr[i] = nextInt();
    		}
    		return arr;
        }
        
        public long[] nextLongArr(int n) {
        	long arr[] = new long[n];
    		for(int i = 0; i < n; i++) {
    			arr[i] = nextLong();
    		}
    		return arr;
        }
        
        public int[] nextIntArr1(int n) {
        	int arr[] = new int[n + 1];
    		for(int i = 1; i <= n; i++) {
    			arr[i] = nextInt();
    		}
    		return arr;
        }
        
        public long[] nextLongArr1(int n) {
        	long arr[] = new long[n + 1];
    		for(int i = 1; i <= n; i++) {
    			arr[i] = nextLong();
    		}
    		return arr;
        }
       
        public void close() {
        	try {
        		if(reader != null) {
        			reader.close();
        		}
        	}
        	catch(Exception e) {
        		
        	}
        	
        	
        }
    }

}