Problem: Takahashi has decided to make a Christmas Tree for the Christmas party in AtCoder, Inc.

A Christmas Tree is a tree with N vertices numbered 1 through N and N-1 edges, whose i-th edge (1\leq i\leq N-1) connects Vertex a_i and b_i.

He would like to make one as follows:

* Specify two non-negative integers A and B.
* Prepare A Christmas Paths whose lengths are at most B. Here, a Christmas Path of length X is a graph with X+1 vertices and X edges such that, if we properly number the vertices 1 through X+1, the i-th edge (1\leq i\leq X) will connect Vertex i and i+1.
* Repeat the following operation until he has one connected tree:
* Select two vertices x and y that belong to different connected components. Combine x and y into one vertex. More precisely, for each edge (p,y) incident to the vertex y, add the edge (p,x). Then, delete the vertex y and all the edges incident to y.
* Properly number the vertices in the tree.



Takahashi would like to find the lexicographically smallest pair (A,B) such that he can make a Christmas Tree, that is, find the smallest A, and find the smallest B under the condition that A is minimized.

Solve this problem for him.

Constraints

* 2 \leq N \leq 10^5
* 1 \leq a_i,b_i \leq N
* The given graph is a tree.

Input

Input is given from Standard Input in the following format:


N
a_1 b_1
:
a_{N-1} b_{N-1}


Output

For the lexicographically smallest (A,B), print A and B with a space in between.

Examples

Input

7
1 2
2 3
2 4
4 5
4 6
6 7


Output

3 2


Input

8
1 2
2 3
3 4
4 5
5 6
5 7
5 8


Output

2 5


Input

10
1 2
2 3
3 4
2 5
6 5
6 7
7 8
5 9
10 5


Output

3 4

Solution (in JAVA):

Solution:
import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.util.Arrays;
import java.util.Random;
import java.util.ArrayList;
import java.util.Map;
import java.io.OutputStreamWriter;
import java.io.OutputStream;
import java.io.PrintStream;
import java.io.IOException;
import java.io.UncheckedIOException;
import java.util.List;
import java.util.AbstractMap;
import java.util.TreeMap;
import java.io.Closeable;
import java.io.Writer;
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 << 27);
        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);
            FChristmasTree solver = new FChristmasTree();
            solver.solve(1, in, out);
            out.close();
        }
    }

    static class FChristmasTree {
        Debug debug = new Debug(false);
        boolean valid;

        public void solve(int testNumber, FastInput in, FastOutput out) {
            int n = in.readInt();
            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);
            }

            for (int i = 0; i < n; i++) {
                nodes[i].children = new IntegerList(nodes[i].adj.size());
            }

            int A = blockNeed(nodes[0], null);

            IntBinarySearch ibs = new IntBinarySearch() {

                public boolean check(int mid) {
                    valid = true;
                    dfs(nodes[0], null, mid);
                    return valid;
                }
            };

            int B = ibs.binarySearch(0, n);

            out.append(A).append(' ').append(B);
        }

        public int blockNeed(Node root, Node p) {
            int child = 0;
            int ans = 0;
            for (Node node : root.adj) {
                if (node == p) {
                    continue;
                }
                child++;
                ans += blockNeed(node, root);
            }
            ans += child / 2;
            if (child % 2 == 1 && p == null) {
                ans++;
            }
            root.odd = p == null && child % 2 == 1 || p != null && child % 2 == 0;
            return ans;
        }

        private void add(TreeMap<Integer, Integer> map, Integer key) {
            map.put(key, map.getOrDefault(key, 0) + 1);
        }

        private void remove(TreeMap<Integer, Integer> map, Integer key) {
            int cnt = map.get(key);
            cnt--;
            if (cnt > 0) {
                map.put(key, cnt);
            } else {
                map.remove(key);
            }
        }

        public int dfs(Node root, Node p, int limit) {
            if (!valid) {
                return 0;
            }

            root.children.clear();
            for (Node node : root.adj) {
                if (node == p) {
                    continue;
                }
                root.children.add(dfs(node, root, limit) + 1);
            }

            if (!valid) {
                return 0;
            }
            root.children.sort();
            int[] data = root.children.getData();
            int n = root.children.size();

            debug.debug("root", root);
            //special
            if (root.odd && p != null) {
                int l = 0;
                int r = n - 1;
                boolean skip = true;
                while (l < r && skip) {
                    if (data[l] + data[r] > limit) {
                        skip = false;
                    }
                    l++;
                    r--;
                }

                if (skip) {
                    return 0;
                }
            }

            if (root.odd && n > 0) {
                if (data[n - 1] > limit) {
                    valid = false;
                    return 0;
                }
                n--;
            }

            TreeMap<Integer, Integer> map = new TreeMap<>();
            for (int i = 0; i < n; i++) {
                add(map, data[i]);
            }

            Integer ret = null;
            while (!map.isEmpty()) {
                Integer max = map.lastKey();
                remove(map, max);
                Integer floor = map.floorKey(limit - max);
                if (floor == null) {
                    if (ret == null) {
                        ret = max;
                        continue;
                    }
                    valid = false;
                    return 0;
                }
                remove(map, floor);
            }
            return ret == null ? 0 : ret;
        }

    }

    static class DigitUtils {
        private DigitUtils() {
        }

        public static int average(int a, int b) {
            int ans = (a / 2) + (b / 2);
            switch (a % 2 + b % 2) {
                case 2:
                    ans++;
                    break;
                case -1:
                case -2:
                    ans--;
                    break;
            }
            return ans;
        }

    }

    static class SequenceUtils {
        public static boolean equal(int[] a, int al, int ar, int[] b, int bl, int br) {
            if ((ar - al) != (br - bl)) {
                return false;
            }
            for (int i = al, j = bl; i <= ar; i++, j++) {
                if (a[i] != b[j]) {
                    return false;
                }
            }
            return true;
        }

    }

    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;
        }

    }

    static class Debug {
        private boolean offline;
        private PrintStream out = System.err;
        static int[] empty = new int[0];

        public Debug(boolean enable) {
            offline = enable && System.getSecurityManager() == null;
        }

        public Debug debug(String name, Object x) {
            return debug(name, x, empty);
        }

        public Debug debug(String name, Object x, int... indexes) {
            if (offline) {
                if (x == null || !x.getClass().isArray()) {
                    out.append(name);
                    for (int i : indexes) {
                        out.printf("[%d]", i);
                    }
                    out.append("=").append("" + x);
                    out.println();
                } else {
                    indexes = Arrays.copyOf(indexes, indexes.length + 1);
                    if (x instanceof byte[]) {
                        byte[] arr = (byte[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    } else if (x instanceof short[]) {
                        short[] arr = (short[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    } else if (x instanceof boolean[]) {
                        boolean[] arr = (boolean[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    } else if (x instanceof char[]) {
                        char[] arr = (char[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    } else if (x instanceof int[]) {
                        int[] arr = (int[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    } else if (x instanceof float[]) {
                        float[] arr = (float[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    } else if (x instanceof double[]) {
                        double[] arr = (double[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    } else if (x instanceof long[]) {
                        long[] arr = (long[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    } else {
                        Object[] arr = (Object[]) x;
                        for (int i = 0; i < arr.length; i++) {
                            indexes[indexes.length - 1] = i;
                            debug(name, arr[i], indexes);
                        }
                    }
                }
            }
            return this;
        }

    }

    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 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 RandomWrapper {
        private Random random;
        public static RandomWrapper INSTANCE = new RandomWrapper(new Random());

        public RandomWrapper() {
            this(new Random());
        }

        public RandomWrapper(Random random) {
            this.random = random;
        }

        public int nextInt(int l, int r) {
            return random.nextInt(r - l + 1) + l;
        }

    }

    static class Randomized {
        public static void shuffle(int[] data, int from, int to) {
            to--;
            for (int i = from; i <= to; i++) {
                int s = nextInt(i, to);
                int tmp = data[i];
                data[i] = data[s];
                data[s] = tmp;
            }
        }

        public static int nextInt(int l, int r) {
            return RandomWrapper.INSTANCE.nextInt(l, r);
        }

    }

    static class Node {
        List<Node> adj = new ArrayList<>();
        IntegerList children;
        int id;
        boolean odd;

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

    }

    static abstract class IntBinarySearch {
        public abstract boolean check(int mid);

        public int binarySearch(int l, int r) {
            if (l > r) {
                throw new IllegalArgumentException();
            }
            while (l < r) {
                int mid = DigitUtils.average(l, r);
                if (check(mid)) {
                    r = mid;
                } else {
                    l = mid + 1;
                }
            }
            return l;
        }

    }

    static class IntegerList implements Cloneable {
        private int size;
        private int cap;
        private int[] data;
        private static final int[] EMPTY = new int[0];

        public int[] getData() {
            return data;
        }

        public IntegerList(int cap) {
            this.cap = cap;
            if (cap == 0) {
                data = EMPTY;
            } else {
                data = new int[cap];
            }
        }

        public IntegerList(IntegerList list) {
            this.size = list.size;
            this.cap = list.cap;
            this.data = Arrays.copyOf(list.data, size);
        }

        public IntegerList() {
            this(0);
        }

        public void ensureSpace(int req) {
            if (req > cap) {
                while (cap < req) {
                    cap = Math.max(cap + 10, 2 * cap);
                }
                data = Arrays.copyOf(data, cap);
            }
        }

        public void add(int x) {
            ensureSpace(size + 1);
            data[size++] = x;
        }

        public void addAll(int[] x, int offset, int len) {
            ensureSpace(size + len);
            System.arraycopy(x, offset, data, size, len);
            size += len;
        }

        public void addAll(IntegerList list) {
            addAll(list.data, 0, list.size);
        }

        public void sort() {
            if (size <= 1) {
                return;
            }
            Randomized.shuffle(data, 0, size);
            Arrays.sort(data, 0, size);
        }

        public int size() {
            return size;
        }

        public int[] toArray() {
            return Arrays.copyOf(data, size);
        }

        public void clear() {
            size = 0;
        }

        public String toString() {
            return Arrays.toString(toArray());
        }

        public boolean equals(Object obj) {
            if (!(obj instanceof IntegerList)) {
                return false;
            }
            IntegerList other = (IntegerList) obj;
            return SequenceUtils.equal(data, 0, size - 1, other.data, 0, other.size - 1);
        }

        public int hashCode() {
            int h = 1;
            for (int i = 0; i < size; i++) {
                h = h * 31 + Integer.hashCode(data[i]);
            }
            return h;
        }

        public IntegerList clone() {
            IntegerList ans = new IntegerList();
            ans.addAll(this);
            return ans;
        }

    }
}