Problem: You are given an integer k and a tree T with n nodes (n is even).

Let dist(u, v) be the number of edges on the shortest path from node u to node v in T.

Let us define a undirected weighted complete graph G = (V, E) as following: 

  * V = \\{x ∣ 1 ≤ x ≤ n \} i.e. the set of integers from 1 to n
  * E = \{(u, v, w) ∣ 1 ≤ u, v ≤ n, u ≠ v, w = dist(u, v) \} i.e. there is an edge between every pair of distinct nodes, the weight being the distance between their respective nodes in T 



Your task is simple, find a perfect matching in G with total edge weight k (1 ≤ k ≤ n^2).

Input

The first line of input contains two integers n, k (2 ≤ n ≤ 100 000, n is even, 1 ≤ k ≤ n^2): number of nodes and the total edge weight of the perfect matching you need to find.

The i-th of the following n - 1 lines contains two integers v_i, u_i (1 ≤ v_i, u_i ≤ n) denoting an edge between v_i and u_i in T. It is guaranteed that the given graph is a tree. 

Output

If there are no matchings that satisfy the above condition, output "NO" (without quotes) on a single line. 

Otherwise, you should output "YES" (without quotes) on the first line of output.

You should then output n/2 lines, the i-th line containing p_i, q_i (1 ≤ p_i, q_i ≤ n): the i-th pair of the matching.

Examples

Input


4 2
1 2
2 3
3 4


Output


YES
2 1
3 4


Input


4 4
1 2
2 3
3 4


Output


YES
3 1
2 4

Note

A tree is a connected acyclic undirected graph.

A matching is set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share a common vertex. 

A perfect matching is a matching which matches all vertices of the graph; that is, every vertex of the graph is incident to exactly one edge of the matching.

Solution (in JAVA):

Solution:
import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.util.PriorityQueue;
import java.util.AbstractQueue;
import java.io.IOException;
import java.util.TreeSet;
import java.util.ArrayList;
import java.io.UncheckedIOException;
import java.util.List;
import java.util.AbstractCollection;
import java.io.Closeable;
import java.io.Writer;
import java.io.OutputStreamWriter;
import java.util.Comparator;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 */
public class Main {
    public static void main(String[] args) throws Exception {
        Thread thread = new Thread(null, new TaskAdapter(), "", 1 << 29);
        thread.start();
        thread.join();
    }

    static class TaskAdapter implements Runnable {
        @Override
        public void run() {
            InputStream inputStream = System.in;
            OutputStream outputStream = System.out;
            FastInput in = new FastInput(inputStream);
            FastOutput out = new FastOutput(outputStream);
            EDistanceMatching solver = new EDistanceMatching();
            solver.solve(1, in, out);
            out.close();
        }
    }

    static class EDistanceMatching {
        public void solve(int testNumber, FastInput in, FastOutput out) {
            int n = in.readInt();
            long k = in.readLong();
            Node[] nodes = new Node[n];
            for (int i = 0; i < n; i++) {
                nodes[i] = new Node();
                nodes[i].id = i;
            }
            for (int i = 0; i < n - 1; i++) {
                Node a = nodes[in.readInt() - 1];
                Node b = nodes[in.readInt() - 1];
                a.adj.add(b);
                b.adj.add(a);
            }

            dfsForSize(nodes[0], null);
            Node root = dfsForCentroid(nodes[0], null, n);
            dfsForDepth(root, null);
            dfsForSize(root, null);

            long sumOfDepth = 0;
            for (Node node : nodes) {
                sumOfDepth += node.depth;
            }

            long max = sumOfDepth;
            long min = 0;
            for (Node node : nodes) {
                min += node.size % 2;
            }

            if (min % 2 != k % 2 || min > k || k > max) {
                out.println("NO");
                return;
            }
            out.println("YES");

            long cur = max;
            long target = k;
            PriorityQueue<TreeSet<Node>> pq = new PriorityQueue<>((a, b) -> -Integer.compare(a.size(), b.size()));
            for (Node node : root.adj) {
                TreeSet<Node> set = new TreeSet<>(Node.sortByDepth);
                collect(node, root, set);
                pq.add(set);
            }

            while (cur > target) {
                TreeSet<Node> top = pq.remove();
                Node leaf = top.last();
                leaf = leaf.p;
                while (cur - leaf.depth * 2 < target) {
                    leaf = leaf.p;
                }

                cur -= leaf.depth * 2;
                Node a = pop(leaf.adj);
                Node b = pop(leaf.adj);
                if (b == null) {
                    b = leaf;
                }
                a.mate = b;
                b.mate = a;
                top.remove(a);
                top.remove(b);

                if (!top.isEmpty()) {
                    pq.add(top);
                }
            }

            TreeSet<Node> rootSet = new TreeSet<>(Node.sortByDepth);
            rootSet.add(root);
            pq.add(rootSet);
            while (!pq.isEmpty()) {
                TreeSet<Node> a = pq.remove();
                TreeSet<Node> b = pq.remove();
                Node u = a.pollFirst();
                Node v = b.pollFirst();
                u.mate = v;
                v.mate = u;

                if (!a.isEmpty()) {
                    pq.add(a);
                }
                if (!b.isEmpty()) {
                    pq.add(b);
                }
            }

            for (Node node : nodes) {
                if (node.id > node.mate.id) {
                    continue;
                }
                out.append(node.id + 1).append(' ').append(node.mate.id + 1).println();
            }
        }

        Node pop(List<Node> list) {
            while (true) {
                if (list.isEmpty()) {
                    return null;
                }
                Node ans = list.remove(list.size() - 1);
                if (ans.mate == null) {
                    return ans;
                }
            }
        }

        public void collect(Node root, Node p, TreeSet<Node> trace) {
            root.p = p;
            root.adj.remove(root.p);
            for (Node node : root.adj) {
                if (node == p) {
                    continue;
                }
                collect(node, root, trace);
            }
            trace.add(root);
        }

        public void dfsForDepth(Node root, Node p) {
            root.depth = p == null ? 0 : p.depth + 1;
            for (Node node : root.adj) {
                if (node == p) {
                    continue;
                }
                dfsForDepth(node, root);
            }
        }

        public void dfsForSize(Node root, Node p) {
            root.size = 1;
            for (Node node : root.adj) {
                if (node == p) {
                    continue;
                }
                dfsForSize(node, root);
                root.size += node.size;
            }
        }

        public Node dfsForCentroid(Node root, Node p, int n) {
            int maxSub = n - root.size;
            for (Node node : root.adj) {
                if (node == p) {
                    continue;
                }
                Node ans = dfsForCentroid(node, root, n);
                if (ans != null) {
                    return ans;
                }
                maxSub = Math.max(maxSub, node.size);
            }
            return maxSub * 2 <= n ? root : null;
        }

    }

    static class FastOutput implements AutoCloseable, Closeable, Appendable {
        private StringBuilder cache = new StringBuilder(10 << 20);
        private final Writer os;

        public FastOutput append(CharSequence csq) {
            cache.append(csq);
            return this;
        }

        public FastOutput append(CharSequence csq, int start, int end) {
            cache.append(csq, start, end);
            return this;
        }

        public FastOutput(Writer os) {
            this.os = os;
        }

        public FastOutput(OutputStream os) {
            this(new OutputStreamWriter(os));
        }

        public FastOutput append(char c) {
            cache.append(c);
            return this;
        }

        public FastOutput append(int c) {
            cache.append(c);
            return this;
        }

        public FastOutput append(String c) {
            cache.append(c);
            return this;
        }

        public FastOutput println(String c) {
            return append(c).println();
        }

        public FastOutput println() {
            cache.append(System.lineSeparator());
            return this;
        }

        public FastOutput flush() {
            try {
                os.append(cache);
                os.flush();
                cache.setLength(0);
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
            return this;
        }

        public void close() {
            flush();
            try {
                os.close();
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
        }

        public String toString() {
            return cache.toString();
        }

    }

    static class FastInput {
        private final InputStream is;
        private byte[] buf = new byte[1 << 13];
        private int bufLen;
        private int bufOffset;
        private int next;

        public FastInput(InputStream is) {
            this.is = is;
        }

        private int read() {
            while (bufLen == bufOffset) {
                bufOffset = 0;
                try {
                    bufLen = is.read(buf);
                } catch (IOException e) {
                    bufLen = -1;
                }
                if (bufLen == -1) {
                    return -1;
                }
            }
            return buf[bufOffset++];
        }

        public void skipBlank() {
            while (next >= 0 && next <= 32) {
                next = read();
            }
        }

        public int readInt() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            int val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

        public long readLong() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            long val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

    }

    static class Node {
        List<Node> adj = new ArrayList<>();
        int depth;
        int size;
        Node p;
        Node mate;
        int id;
        public static Comparator<Node> sortByDepth = (a, b) -> a.depth == b.depth ? Integer.compare(a.id, b.id) : Integer.compare(a.depth, b.depth);

        public String toString() {
            return "" + (id + 1);
        }

    }
}