Choosing Subtree is Fun

There is a tree consisting of n vertices. The vertices are numbered from 1 to n.

Let’s define the length of an interval [l, r] as the value r - l + 1. The score of a subtree of this tree is the maximum length of such an interval [l, r] that, the vertices with numbers l, l + 1, …, r belong to the subtree.

Considering all subtrees of the tree whose size is at most k, return the maximum score of the subtree. Note, that in this problem tree is not rooted, so a subtree — is an arbitrary connected subgraph of the tree.

Input

There are two integers in the first line, n and k (1 ≤ k ≤ n ≤ 10⁵). Each of the next n - 1 lines contains integers a _i and b _i (1 ≤ a _i, b _i ≤ n, a _i ≠ b _i). That means a _i and b _i are connected by a tree edge.

It is guaranteed that the input represents a tree.

Output

Output should contain a single integer — the maximum possible score.

Examples

input

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

output

3

input

16 7
13 11
12 11
2 14
8 6
9 15
16 11
5 14
6 15
4 3
11 15
15 14
10 1
3 14
14 7
1 7

output

6

Note

For the first case, there is some subtree whose size is at most 6, including 3 consecutive numbers of vertices. For example, the subtree that consists of {1, 3, 4, 5, 7, 8} or of {1, 4, 6, 7, 8, 10} includes 3 consecutive numbers of vertices. But there is no subtree whose size is at most 6, which includes 4 or more consecutive numbers of vertices.

Solution:

#include <iostream>
#include <iomanip>
#include <cstdio>
#include <set>
#include <vector>
#include <map>
#include <cmath>
#include <algorithm>
#include <memory.h>
#include <string>
#include <cstring>
#include <sstream>
#include <cstdlib>
#include <ctime>
#include <cassert>

using namespace std;

const int N = 400010;

int from[N], to[N], pred[N], last[N], de[N], pv[N];
int T, tin[N], tout[N];
int kw, w[N], pos[N];
int s[N], pr[N], ne[N];
bool was[N];
int plca[22][N];

int n;

void dfs(int v) {
tin[v] = ++T;
was[v] = true;
w[++kw] = v;
int j = last[v];
while (j > 0) {
if (!was[to[j]]) {
de[to[j]] = de[v] + 1;
pv[to[j]] = v;
dfs(to[j]);
}
j = pred[j];
}
tout[v] = ++T;
}

void modify(int x, int v) {
while (x <= n) {
    s[x] += v;
    x = (x | (x - 1)) + 1;
  }
}

int findsum(int x) {
  int v = 0;
  while (x > 0) {
v += s[x];
x &= x - 1;
}
return v;
}

int find_left(int p) {
if (findsum(n) == 0) return p;
if (findsum(p) == 0) p = n;
int u = findsum(p);
int ll = 1, rr = p;
while (ll < rr) {
    int mid = (ll + rr) >> 1;
if (findsum(mid) == u) rr = mid;
else ll = mid + 1;
}
return ll;
}

int find_right(int p) {
if (findsum(n) == 0) return p;
if (findsum(n) - findsum(p) == 0) p = 0;
int u = findsum(p);
int ll = p, rr = n;
while (ll < rr) {
    int mid = (ll + rr) >> 1;
if (findsum(mid) == u) ll = mid + 1;
else rr = mid;
}
return ll;
}

bool anc(int x, int y) {
return (tin[x] <= tin[y] && tout[y] <= tout[x]);
}

int lca(int x, int y) {
  if (anc(x, y)) return x;
  for (int j = 20; j >= 0; j--)
if (!anc(plca[j][x], y)) x = plca[j][x];
return pv[x];
}

int sum;

int get(int x, int y) {
x = w[x];
y = w[y];
int z = lca(x, y);
return de[y] - de[z];
}

void add(int v) {
v = pos[v];
pr[v] = find_left(v);
ne[v] = find_right(v);
pr[ne[v]] = v;
ne[pr[v]] = v;
modify(v, 1);
sum -= get(pr[v], ne[v]);
sum += get(pr[v], v);
sum += get(v, ne[v]);
}

void del(int v) {
v = pos[v];
sum -= get(pr[v], v);
sum -= get(v, ne[v]);
sum += get(pr[v], ne[v]);
pr[ne[v]] = pr[v];
ne[pr[v]] = ne[v];
modify(v, -1);
}

int main() {
int k;
scanf("%d %d", &n, &k);
int m = n - 1;
for (int i = 1; i <= m; i++) {
    scanf("%d %d", from + i, to + i);
    from[i + m] = to[i];
    to[i + m] = from[i];
  }
  for (int i = 1; i <= n; i++) last[i] = 0;
  for (int i = 1; i <= m + m; i++) {
    pred[i] = last[from[i]];
    last[from[i]] = i;
  }
  for (int i = 1; i <= n; i++) was[i] = false;
  kw = 0;
  de[1] = 1;
  T = 0;
  dfs(1);
  pv[1] = 1;
  for (int i = 1; i <= n; i++) plca[0][i] = pv[i];
  for (int j = 1; j <= 20; j++)
    for (int i = 1; i <= n; i++) {
      plca[j][i] = plca[j - 1][plca[j - 1][i]];
    }
  for (int i = 1; i <= n; i++) pos[w[i]] = i;
  for (int i = 1; i <= n; i++) s[i] = 0;
  int R = 0, ans = 1;
  sum = 1;
  for (int L = 1; L < n; L++) {
    while (R + 1 <= n) {
      add(R + 1);
      if (sum <= k) R++; else {
        del(R + 1);
        break;
      }
    }
    if (R - L + 1 > ans) ans = R - L + 1;
del(L);
}
printf("%d\n", ans);
return 0;
}