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 n, a, b, P[200001][40];
vector<int> G[200001], q(200001), Parent(200001), volume(200001);
void Prepare() {
  for (int i = 1; i <= n; i++)
    for (int j = 0; (1 << j) < n; j++)
      j == 0 ? P[i][j] = Parent[i] : P[i][j] = -1;
  for (int j = 1; (1 << j) < n; j++)
    for (int i = 1; i <= n; i++)
      if (P[i][j - 1] != -1) P[i][j] = P[P[i][j - 1]][j - 1];
}
int LCA(int u, int v) {
  if (volume[u] < volume[v]) swap(u, v);
  int k = volume[u] - volume[v];
  while (k > 0) {
    int t = log2(k);
    u = P[u][t];
    k -= (1 << t);
  }
  if (u == v) return u;
  k = log2(n);
  for (int j = k; j >= 0; j--)
    if (P[v][j] != -1 && P[v][j] != P[u][j]) {
      u = P[u][j];
      v = P[v][j];
    }
  return Parent[u];
}
int Len(int u, int v) {
  int w = LCA(u, v);
  return volume[u] - volume[w] + volume[v] - volume[w];
}
int main() {
  cin >> n;
  for (int i = 2; i <= n; i++) {
    cin >> q[i];
    G[q[i]].push_back(i);
    volume[i] = volume[q[i]] + 1;
    Parent[i] = q[i];
  }
  Prepare();
  int d = 1;
  a = 1;
  b = G[1][0];
  cout << d << ' ';
  for (int i = 3; i <= n; i++) {
    int w = i;
    int d1 = Len(a, w), d2 = Len(b, w);
    if (max(d1, d2) > d)
      if (d1 > d2) {
        b = w;
        d = d1;
      } else {
        a = w;
        d = d2;
      }
    cout << d << ' ';
  }
  return 0;
}
