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.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.Collection;
import java.util.InputMismatchException;
import java.io.IOException;
import java.util.Random;
import java.util.ArrayList;
import java.util.NoSuchElementException;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        TaskE2 solver = new TaskE2();
        solver.solve(1, in, out);
        out.close();
    }

    static class TaskE2 {
        int n;
        long s;
        TaskE2.Node[] nodes;
        TaskE2.Edge[] edges;

        void dfs(TaskE2.Node cur, TaskE2.Node par, int depth) {
            cur.depth = depth;
            cur.leavesInSubtree = 0;
            if (cur != nodes[1] && cur.adj.size() == 1) {
                cur.leavesInSubtree = 1;
            }
            for (TaskE2.Edge e : cur.adj) {
                if (e.to != par) {
                    dfs(e.to, cur, depth + 1);
                    cur.leavesInSubtree += e.to.leavesInSubtree;
                }
            }
        }

        public void solve(int testNumber, InputReader in, PrintWriter out) {
            int tt = in.nextInt();
            while (tt-- > 0) {
                n = in.nextInt();
                s = in.nextLong();
                nodes = new TaskE2.Node[n + 1];
                for (int i = 1; i <= n; i++) {
                    nodes[i] = new TaskE2.Node();
                }
                edges = new TaskE2.Edge[n - 1];
                for (int i = 0; i < n - 1; i++) {
                    int u = in.nextInt(), v = in.nextInt(), w = in.nextInt(), c = in.nextInt();
                    TaskE2.Node from = nodes[u];
                    TaskE2.Node to = nodes[v];
                    from.adj.add(new TaskE2.Edge(from, to, w, c));
                    to.adj.add(new TaskE2.Edge(to, from, w, c));
                    edges[i] = new TaskE2.Edge(from, to, w, c);
                }

                dfs(nodes[1], null, 0);

                for (TaskE2.Edge e : edges) {
                    if (e.from.depth > e.to.depth) {
                        TaskE2.Node temp = e.from;
                        e.from = e.to;
                        e.to = temp;
                    }
                }

                long total = 0;
                EzLongArrayList[] opt = new EzLongArrayList[3];
                opt[1] = new EzLongArrayList();
                opt[2] = new EzLongArrayList();
                for (TaskE2.Edge e : edges) {
                    total += (long) e.weight * e.to.leavesInSubtree;
                    for (int w = e.weight; w > 0; w /= 2) {
                        opt[e.cost].add((long) (w - w / 2) * e.to.leavesInSubtree);
                    }
                }

                for (int c = 1; c <= 2; c++) {
                    opt[c].sort();
                    opt[c].reverse();
                }

                long[] p1 = new long[opt[1].size() + 1];
                long[] p2 = new long[opt[2].size() + 1];
                for (int i = 1; i <= opt[1].size(); i++) {
                    p1[i] = opt[1].get(i - 1) + p1[i - 1];
                }
                for (int i = 1; i <= opt[2].size(); i++) {
                    p2[i] = opt[2].get(i - 1) + p2[i - 1];
                }

                long ans = (long) 1e18;
                long need = total - s;
                for (int n1 = 0; n1 <= opt[1].size(); n1++) {
                    long cur = p1[n1];
                    int n2 = lowerBound(p2, need - cur);
                    if (n2 < p2.length) {
                        ans = Math.min(ans, (long) n1 + 2 * n2);
                    }
                }

                out.println(ans);
            }
        }

        int lowerBound(long[] a, long val) {
            int ret = -1;
            for (int jump = a.length; jump >= 1; jump /= 2) {
                while (ret + jump < a.length && a[ret + jump] < val) {
                    ret += jump;
                }
            }
            return ret + 1;
        }

        static class Node {
            ArrayList<TaskE2.Edge> adj;
            int depth;
            int leavesInSubtree;

            Node() {
                adj = new ArrayList<>();
            }

        }

        static class Edge {
            TaskE2.Node from;
            TaskE2.Node to;
            int weight;
            int cost;

            public Edge(TaskE2.Node from, TaskE2.Node to, int weight, int cost) {
                this.from = from;
                this.to = to;
                this.weight = weight;
                this.cost = cost;
            }

        }

    }

    static class InputReader {
        private final InputStream is;
        private final byte[] inbuf = new byte[1024];
        private int lenbuf = 0;
        private int ptrbuf = 0;

        public InputReader(InputStream stream) {
            is = stream;
        }

        private int readByte() {
            if (lenbuf == -1) throw new InputMismatchException();
            if (ptrbuf >= lenbuf) {
                ptrbuf = 0;
                try {
                    lenbuf = is.read(inbuf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (lenbuf <= 0) return -1;
            }
            return inbuf[ptrbuf++];
        }

        public int nextInt() {
            int num = 0, b;
            boolean minus = false;
            while ((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-')) ;
            if (b == '-') {
                minus = true;
                b = readByte();
            }

            while (true) {
                if (b >= '0' && b <= '9') {
                    num = num * 10 + (b - '0');
                } else {
                    return minus ? -num : num;
                }
                b = readByte();
            }
        }

        public long nextLong() {
            long num = 0;
            int b;
            boolean minus = false;
            while ((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-')) ;
            if (b == '-') {
                minus = true;
                b = readByte();
            }

            while (true) {
                if (b >= '0' && b <= '9') {
                    num = num * 10 + (b - '0');
                } else {
                    return minus ? -num : num;
                }
                b = readByte();
            }
        }

    }

    static interface EzLongCollection {
        int size();

        EzLongIterator iterator();

        boolean equals(Object object);

        int hashCode();

        String toString();

    }

    static final class EzLongSort {
        private static final Random rnd = new Random();

        private EzLongSort() {
        }

        private static void randomShuffle(long[] a, int left, int right) {
            for (int i = left; i < right; i++) {
                int j = i + rnd.nextInt(right - i);
                long tmp = a[i];
                a[i] = a[j];
                a[j] = tmp;
            }
        }

        public static void safeArraysSort(long[] a, int left, int right) {
            if (left > right || left < 0 || right > a.length) {
                throw new IllegalArgumentException(
                        "Incorrect range [" + left + ", " + right + ") was specified for sorting, length = " + a.length);
            }
            randomShuffle(a, left, right);
            Arrays.sort(a, left, right);
        }

    }

    static class EzLongArrayList implements EzLongList, EzLongStack {
        private static final int DEFAULT_CAPACITY = 10;
        private static final double ENLARGE_SCALE = 2.0;
        private static final int HASHCODE_INITIAL_VALUE = 0x811c9dc5;
        private static final int HASHCODE_MULTIPLIER = 0x01000193;
        private long[] array;
        private int size;

        public EzLongArrayList() {
            this(DEFAULT_CAPACITY);
        }

        public EzLongArrayList(int capacity) {
            if (capacity < 0) {
                throw new IllegalArgumentException("Capacity must be non-negative");
            }
            array = new long[capacity];
            size = 0;
        }

        public EzLongArrayList(EzLongCollection collection) {
            size = collection.size();
            array = new long[size];
            int i = 0;
            for (EzLongIterator iterator = collection.iterator(); iterator.hasNext(); ) {
                array[i++] = iterator.next();
            }
        }

        public EzLongArrayList(long[] srcArray) {
            size = srcArray.length;
            array = new long[size];
            System.arraycopy(srcArray, 0, array, 0, size);
        }

        public EzLongArrayList(Collection<Long> javaCollection) {
            size = javaCollection.size();
            array = new long[size];
            int i = 0;
            for (long element : javaCollection) {
                array[i++] = element;
            }
        }

        public int size() {
            return size;
        }

        public EzLongIterator iterator() {
            return new EzLongArrayListIterator();
        }

        public boolean add(long element) {
            if (size == array.length) {
                enlarge();
            }
            array[size++] = element;
            return true;
        }

        public long get(int index) {
            if (index < 0 || index >= size) {
                throw new IndexOutOfBoundsException("Index " + index + " is out of range, size = " + size);
            }
            return array[index];
        }

        private void enlarge() {
            int newSize = Math.max(size + 1, (int) (size * ENLARGE_SCALE));
            long[] newArray = new long[newSize];
            System.arraycopy(array, 0, newArray, 0, size);
            array = newArray;
        }

        public boolean equals(Object o) {
            if (this == o) return true;
            if (o == null || getClass() != o.getClass()) return false;
            EzLongArrayList that = (EzLongArrayList) o;

            if (size != that.size) {
                return false;
            }
            for (int i = 0; i < size; i++) {
                if (array[i] != that.array[i]) {
                    return false;
                }
            }
            return true;
        }

        public int hashCode() {
            int hash = HASHCODE_INITIAL_VALUE;
            for (int i = 0; i < size; i++) {
                hash = (hash ^ PrimitiveHashCalculator.getHash(array[i])) * HASHCODE_MULTIPLIER;
            }
            return hash;
        }

        public String toString() {
            StringBuilder sb = new StringBuilder();
            sb.append('[');
            for (int i = 0; i < size; i++) {
                sb.append(array[i]);
                if (i < size - 1) {
                    sb.append(", ");
                }
            }
            sb.append(']');
            return sb.toString();
        }

        public void sort() {
            EzLongSort.safeArraysSort(array, 0, size);
        }

        public void reverse() {
            reverse(0, size);
        }

        public void reverse(int left, int right) {
            if (left > right || right > size) {
                throw new IllegalArgumentException(
                        "Incorrect range [" + left + ", " + right + ") was specified for reversing, length = " + size);
            }
            for (int i = left, j = right - 1; i < j; i++, j--) {
                long temp = array[i];
                array[i] = array[j];
                array[j] = temp;
            }
        }

        private class EzLongArrayListIterator implements EzLongIterator {
            private int curIndex = 0;

            public boolean hasNext() {
                return curIndex < size;
            }

            public long next() {
                if (curIndex == size) {
                    throw new NoSuchElementException("Iterator doesn't have more elements");
                }
                return array[curIndex++];
            }

        }

    }

    static interface EzLongIterator {
        boolean hasNext();

        long next();

    }

    static interface EzLongList extends EzLongCollection {
        int size();

        EzLongIterator iterator();

        boolean equals(Object object);

        int hashCode();

        String toString();

    }

    static interface EzLongStack extends EzLongCollection {
        int size();

        EzLongIterator iterator();

        boolean equals(Object object);

        int hashCode();

        String toString();

    }

    static final class PrimitiveHashCalculator {
        private PrimitiveHashCalculator() {
        }

        public static int getHash(long x) {
            return (int) x ^ (int) (x >>> 32);
        }

    }
}