Breaking news from zombie neurology! It turns out that – contrary to previous beliefs – every zombie is born with a single brain, and only later it evolves into a complicated brain structure. In fact, whenever a zombie consumes a brain, a new brain appears in its nervous system and gets immediately connected to one of the already existing brains using a single brain connector. Researchers are now interested in monitoring the brain latency of a zombie. Your task is to write a program which, given a history of evolution of a zombie's nervous system, computes its brain latency at every stage.

Input

The first line of the input contains one number n – the number of brains in the final nervous system (2 ≤ n ≤ 200000). In the second line a history of zombie's nervous system evolution is given. For convenience, we number all the brains by 1, 2, ..., n in the same order as they appear in the nervous system (the zombie is born with a single brain, number 1, and subsequently brains 2, 3, ..., n are added). The second line contains n - 1 space-separated numbers p2, p3, ..., pn, meaning that after a new brain k is added to the system, it gets connected to a parent-brain <image>.

Output

Output n - 1 space-separated numbers – the brain latencies after the brain number k is added, for k = 2, 3, ..., n.

Example

Input

6
1
2
2
1
5


Output

1 2 2 3 4 
#include <bits/stdc++.h>
using namespace std;
bool debug = 0;
int n, m, k;
int dx[4] = {0, 1, 0, -1}, dy[4] = {1, 0, -1, 0};
string direc = "RDLU";
long long ln, lk, lm;
void etp(bool f = 0) {
  puts(f ? "yes" : "no");
  exit(0);
}
void addmod(int &x, int y, int mod = 1000000007) {
  x += y;
  if (x >= mod) x -= mod;
}
void et() {
  puts("-1");
  exit(0);
}
vector<int> mp[200105];
int p[200105][20], f[200105], d[200105];
void dfs(int x, int pa) {
  d[x] = d[pa] + 1;
  p[x][0] = pa;
  for (int c : mp[x]) dfs(c, x);
}
void init() {
  for (int j = 1; (1 << j) <= n; j++)
    for (int i = 1; i <= n; i++) p[i][j] = p[p[i][j - 1]][j - 1];
}
int lca(int a, int b) {
  if (d[a] > d[b]) swap(a, b);
  int f = d[b] - d[a];
  for (int i = 0; (1 << i) <= f; i++)
    if ((1 << i) & f) b = p[b][i];
  if (a != b) {
    for (int i = 19; i >= 0; i--)
      if (p[a][i] != p[b][i]) {
        a = p[a][i];
        b = p[b][i];
      }
    a = p[a][0];
  }
  return a;
}
int cal(int u, int v) {
  int c = lca(u, v);
  return d[u] + d[v] - d[c] * 2;
}
void fmain() {
  scanf("%d", &n);
  for (int i = 2; i <= n; i++) {
    scanf("%d", f + i);
    mp[f[i]].push_back(i);
  }
  dfs(1, 0);
  init();
  vector<pair<int, pair<int, int>>> vp;
  vp.push_back({0, {1, 1}});
  for (int i = 2; i <= n; i++) {
    int u = vp[0].second.first, v = vp[0].second.second;
    vp.push_back({-cal(u, i), {u, i}});
    vp.push_back({-cal(i, v), {v, i}});
    sort(vp.begin(), vp.end());
    vp.pop_back();
    vp.pop_back();
    printf("%d ", -vp[0].first);
  }
}
int main() {
  fmain();
  return 0;
}
