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;
int J[200000][19];
int ent[200000];
int lev[200000];
int tick;
int hl[200000];
void dfs(int i, vector<vector<int> >& vex) {
  ent[i] = ++tick;
  for (int k = 1; k < 19; ++k) J[i][k] = J[J[i][k - 1]][k - 1];
  for (int& r : vex[i]) {
    J[r][0] = i;
    hl[r] = hl[i] + 1;
    dfs(r, vex);
  }
  lev[i] = ++tick;
  return;
}
bool anc(int i, int j) { return ent[i] <= ent[j] && lev[j] <= lev[i]; }
int lca(int i, int j) {
  if (anc(i, j)) return i;
  if (anc(j, i)) return j;
  for (int k = 18; k >= 0; --k)
    if (!anc(J[i][k], j)) i = J[i][k];
  return J[i][0];
}
int dist(int i, int j) { return hl[i] + hl[j] - 2 * hl[lca(i, j)]; }
int main() {
  ios::sync_with_stdio(false);
  cin.tie(NULL);
  int n;
  cin >> n;
  int u;
  vector<vector<int> > vex(n);
  pair<int, int> mn(0, 0);
  for (int i = 1; i < n; ++i) {
    cin >> u;
    vex[u - 1].push_back(i);
  }
  dfs(0, vex);
  for (int i = 1; i < n; ++i) {
    int fi = dist(mn.first, i);
    int si = dist(i, mn.second);
    int fs = dist(mn.first, mn.second);
    if (fi > si && fi > fs)
      mn.second = i;
    else if (si > fs)
      mn.first = i;
    cout << dist(mn.first, mn.second) << ' ';
  }
  cout << endl;
  return 0;
}
