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;
const int MAXN = 2e5 + 1;
int n;
int L[MAXN][18];
vector<int> A[MAXN];
int dist[MAXN];
int sol[MAXN];
void dfs(int nod) {
  for (int i = 0; i < A[nod].size(); i++) {
    int v = A[nod][i];
    dist[v] = dist[nod] + 1;
    dfs(v);
  }
}
int lca_query(int a, int b) {
  if (dist[a] > dist[b]) swap(a, b);
  for (int e = 17; e > -1; e--) {
    if ((1 << e) <= dist[b] - dist[a]) b = L[b][e];
  }
  if (a == b) return a;
  for (int e = 17; e >= 0; e--) {
    if (L[b][e] != L[a][e]) {
      b = L[b][e];
      a = L[a][e];
    }
  }
  return L[a][0];
}
int dist_query(int a, int b) {
  int lca = lca_query(a, b);
  int d = dist[a] + dist[b] - 2 * dist[lca];
  return d;
}
int main() {
  cin >> n;
  for (int i = 2; i <= n; i++) {
    int p;
    cin >> p;
    L[i][0] = p;
    A[p].push_back(i);
  }
  for (int e = 1; (1 << e) < n; e++)
    for (int j = 1; j <= n; j++) L[j][e] = L[L[j][e - 1]][e - 1];
  dfs(1);
  int v, u;
  v = 1;
  u = 2;
  sol[2] = 1;
  cout << 1;
  for (int i = 3; i <= n; i++) {
    sol[i] = sol[i - 1];
    int d1 = dist_query(i, u);
    int vv = v, uu = u;
    if (sol[i] < d1) {
      vv = i;
      sol[i] = d1;
    }
    int d2 = dist_query(i, v);
    if (sol[i] < d2) {
      uu = i;
      sol[i] = d2;
    }
    v = vv;
    u = uu;
    cout << ' ' << sol[i];
  }
  return 0;
}
