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 nmax = 200005;
const long long mod = 1e6 + 3;
const long long inf = 1e9;
const long double EPS = 1e-6;
const long double PI = acos(-1.0);
const int kor = 320;
vector<vector<int> > g;
int up[nmax][20];
int d[nmax];
int p[nmax];
int l;
int tt;
int tin[nmax], tout[nmax];
void dfs(int v, int batia) {
  up[v][0] = batia;
  tin[v] = tt++;
  for (int i = 1; i <= l; ++i) up[v][i] = up[up[v][i - 1]][i - 1];
  for (int j = 0; j < g[v].size(); ++j) {
    int to = g[v][j];
    if (to != batia) dfs(to, v);
  }
  tout[v] = tt++;
}
int par(int a, int b) { return (tin[a] <= tin[b] && tout[a] >= tout[b]); }
int lca(int a, int b) {
  if (par(a, b)) return a;
  if (par(b, a)) return b;
  for (int i = l; i >= 0; --i) {
    if (!par(up[a][i], b)) a = up[a][i];
  }
  return up[a][0];
}
int main() {
  ios_base::sync_with_stdio(false);
  int n;
  cin >> n;
  g.assign(n + 1, vector<int>());
  for (int i = 2; i <= n; ++i) {
    int x;
    cin >> x;
    d[i] = d[x] + 1;
    g[x].push_back(i);
  }
  l = 1;
  while ((1 << l) <= n) l++;
  dfs(1, 1);
  int u = 1, v = 1;
  int an = 0;
  vector<int> w;
  for (int i = 2; i <= n; ++i) {
    int f = lca(u, i);
    f = d[i] + d[u] - d[f] - d[f];
    int s = lca(v, i);
    s = d[i] + d[v] - d[s] - d[s];
    if (f >= s) {
      if (f > an) {
        v = i;
        an = f;
      }
    } else {
      if (s > an) {
        u = i;
        an = s;
      }
    }
    w.push_back(an);
  }
  for (int i = 0; i < n - 1; ++i) cout << w[i] << ' ';
  cout << '\n';
  return 0;
}
