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:
//package Codeforces.Round661Div3;

import java.io.*;
import java.util.*;

public class E2_WeightsDivision_Hard {
    static class Node{
        int id;
        boolean isVisited;
        ArrayList<Edge> edgelist = new ArrayList<>();
        Node(int id){
            this.id = id;
            this.isVisited = false;
        }
    }
    static class Edge implements Comparable<Edge>{
        Node u;
        Node v;
        long weight;
        long weightBy2;
        int cost;
        long leafs_factor;
        Edge(Node u, Node v, long weight, int cost){
            this.u = u;
            this.v = v;
            this.weight = weight;
            this.weightBy2 = this.weight/2;
            this.cost = cost;
            this.leafs_factor = 0;
        }
        public int compareTo(Edge e){
            return Long.compare(this.weight-this.weightBy2, e.weight-e.weightBy2);
        }
        public String toString(){
            return this.u.id+" "+this.v.id+" "+this.weight+" "+this.weightBy2+" "+this.cost+" "+this.leafs_factor;
        }
    }
    static class Tree{
        ArrayList<Node> nodelist;
        ArrayList<Edge> edgelist;
        Tree(int n){
            nodelist = new ArrayList<>(n);
            edgelist = new ArrayList<>(n-1);
            for(int i=0;i<n;i++){
                nodelist.add(new Node(i));
            }
        }
        public void addEdge(int u, int v, long weight, int cost){
            Node nu = nodelist.get(u);
            Node nv = nodelist.get(v);
            Edge e = new Edge(nu, nv, weight, cost);

            nu.edgelist.add(e);
            nv.edgelist.add(e);
            edgelist.add(e);
        }
        public int dfs(Node source){
            source.isVisited = true;
            if(source.edgelist.size()==1 && source.id!=0)
                return 1;

            for(Edge e: source.edgelist){
                Node child;
                if(e.u==source){
                    child = e.v;
                }
                else child = e.u;

                if(!child.isVisited) {
                    e.leafs_factor = dfs(child);
                }
            }

            int sum = 0;
            for(Edge e: source.edgelist){
                sum += e.leafs_factor;
            }
            return sum;
        }
    }
    static int binarySearch(ArrayList<Long> list, long key, int ll, int ul){
        if(ll<=ul){
            int mid = (ll+ul)/2;
            if(list.get(mid)==key){
                return mid;
            }
            else if(list.get(mid)<key){
                if(mid==0){
                    return mid;
                }
                else if(list.get(mid-1)>key){
                    return mid;
                }
                else return binarySearch(list, key, ll, mid-1);
            }
            else{
                if(mid==list.size()-1){
                    return -1;
                }
                else return binarySearch(list, key, mid+1, ul);
            }
        }
        else return -1;
    }
    public static void main(String[] args) throws IOException {
        Soumit sc = new Soumit();

        int t = sc.nextInt();
        StringBuilder sb = new StringBuilder();
        while (t-->0){
            int n = sc.nextInt();
            long k = sc.nextLong();

            Tree tr = new Tree(n);
            for(int i=0;i<n-1;i++){
                int u = sc.nextInt()-1;
                int v = sc.nextInt()-1;
                long weight = sc.nextLong();
                int cost = sc.nextInt();
                tr.addEdge(u, v, weight, cost);
            }

            tr.dfs(tr.nodelist.get(0));
            /*for(Edge e: tr.edgelist){
                System.out.println(e);
            }*/

            long totsum1 = 0;
            long totsum2 = 0;
            PriorityQueue<Edge> heap1 = new PriorityQueue<>(Collections.reverseOrder());
            PriorityQueue<Edge> heap2 = new PriorityQueue<>(Collections.reverseOrder());
            for(Edge e: tr.edgelist){
                e.weight *= e.leafs_factor;
                e.weightBy2 *= e.leafs_factor;
                if(e.cost==1){
                    totsum1 += e.weight;
                    heap1.add(e);
                }
                else{
                    totsum2 += e.weight;
                    heap2.add(e);
                }
            }

            ArrayList<Long> list1 = new ArrayList<>();
            list1.add(totsum1);
            ArrayList<Long> list2 = new ArrayList<>();
            list2.add(totsum2);
            while(!heap1.isEmpty()){
                Edge e = heap1.remove();
                totsum1 -= (e.weight-e.weightBy2);
                e.weight = e.weightBy2;
                e.weightBy2 /= e.leafs_factor;
                e.weightBy2 /= 2;
                e.weightBy2 *= e.leafs_factor;

                if(e.weight!=0){
                    heap1.add(e);
                }
                list1.add(totsum1);
            }

            while(!heap2.isEmpty()){
                Edge e = heap2.remove();
                totsum2 -= (e.weight-e.weightBy2);
                e.weight = e.weightBy2;
                e.weightBy2 /= e.leafs_factor;
                e.weightBy2 /= 2;
                e.weightBy2 *= e.leafs_factor;

                if(e.weight!=0){
                    heap2.add(e);
                }
                list2.add(totsum2);
            }

            /*System.out.println(list1);
            System.out.println(list2);*/

            int cost = Integer.MAX_VALUE;
            for(int i=0;i<list1.size();i++){
                long lval1 = list1.get(i);
                if(lval1>k){
                    continue;
                }

                int li2 = binarySearch(list2, k-lval1, 0, list2.size()-1);
                //System.out.println(i+" "+li2);
                if(li2!=-1)
                    cost = Math.min(cost, i+li2*2);
            }

            sb.append(cost).append("\n");
        }

        System.out.println(sb);

        sc.close();
    }

    static class Soumit {
        final private int BUFFER_SIZE = 1 << 18;
        final private DataInputStream din;
        final private byte[] buffer;
        private PrintWriter pw;
        private int bufferPointer, bytesRead;
        StringTokenizer st;

        public Soumit() {
            din = new DataInputStream(System.in);
            buffer = new byte[BUFFER_SIZE];
            bufferPointer = bytesRead = 0;
        }

        public Soumit(String file_name) throws IOException {
            din = new DataInputStream(new FileInputStream(file_name));
            buffer = new byte[BUFFER_SIZE];
            bufferPointer = bytesRead = 0;
        }

        public void streamOutput(String file) throws IOException {
            FileWriter fw = new FileWriter(file);
            BufferedWriter bw = new BufferedWriter(fw);
            pw = new PrintWriter(bw);
        }

        public void println(String a) {
            pw.println(a);
        }

        public void print(String a) {
            pw.print(a);
        }

        public String readLine() throws IOException {
            byte[] buf = new byte[3000064]; // line length
            int cnt = 0, c;
            while ((c = read()) != -1) {
                if (c == '\n')
                    break;
                buf[cnt++] = (byte) c;
            }
            return new String(buf, 0, cnt);
        }

        String next() {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(readLine());
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

        public void sort(int[] arr) {
            ArrayList<Integer> arlist = new ArrayList<>();
            for (int i : arr)
                arlist.add(i);

            Collections.sort(arlist);
            for (int i = 0; i < arr.length; i++)
                arr[i] = arlist.get(i);
        }

        public void sort(long[] arr) {
            ArrayList<Long> arlist = new ArrayList<>();
            for (long i : arr)
                arlist.add(i);

            Collections.sort(arlist);
            for (int i = 0; i < arr.length; i++)
                arr[i] = arlist.get(i);
        }

        public int[] nextIntArray(int n) throws IOException {
            int[] arr = new int[n];
            for (int i = 0; i < n; i++) {
                arr[i] = nextInt();
            }

            return arr;
        }

        public long[] nextLongArray(int n) throws IOException {
            long[] arr = new long[n];
            for (int i = 0; i < n; i++) {
                arr[i] = nextLong();
            }

            return arr;
        }

        public double[] nextDoubleArray(int n) throws IOException {
            double[] arr = new double[n];
            for (int i = 0; i < n; i++) {
                arr[i] = nextDouble();
            }

            return arr;
        }

        public int nextInt() throws IOException {
            int ret = 0;
            byte c = read();
            while (c <= ' ')
                c = read();
            boolean neg = (c == '-');
            if (neg)
                c = read();
            do {
                ret = ret * 10 + c - '0';
            } while ((c = read()) >= '0' && c <= '9');

            if (neg)
                return -ret;
            return ret;
        }

        public long nextLong() throws IOException {
            long ret = 0;
            byte c = read();
            while (c <= ' ')
                c = read();
            boolean neg = (c == '-');
            if (neg)
                c = read();
            do {
                ret = ret * 10 + c - '0';
            }
            while ((c = read()) >= '0' && c <= '9');
            if (neg)
                return -ret;
            return ret;
        }

        public double nextDouble() throws IOException {
            double ret = 0, div = 1;
            byte c = read();
            while (c <= ' ')
                c = read();
            boolean neg = (c == '-');
            if (neg)
                c = read();

            do {
                ret = ret * 10 + c - '0';
            }
            while ((c = read()) >= '0' && c <= '9');

            if (c == '.') {
                while ((c = read()) >= '0' && c <= '9') {
                    ret += (c - '0') / (div *= 10);
                }
            }

            if (neg)
                return -ret;
            return ret;
        }

        private void fillBuffer() throws IOException {
            bytesRead = din.read(buffer, bufferPointer = 0, BUFFER_SIZE);
            if (bytesRead == -1)
                buffer[0] = -1;
        }

        private byte read() throws IOException {
            if (bufferPointer == bytesRead)
                fillBuffer();
            return buffer[bufferPointer++];
        }

        public void close() throws IOException {
            /*if (din == null)
                return;*/
            if (din != null) din.close();
            if (pw != null) pw.close();
        }
    }
}