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 + 10, M = 20;
int par[M][maxn], d[maxn];
vector<int> adj[maxn];
void dfs(int v, int p) {
  par[0][v] = p;
  for (int i = 1; i < M; i++) par[i][v] = par[i - 1][par[i - 1][v]];
  for (int u : adj[v]) {
    if (u == p) continue;
    d[u] = d[v] + 1;
    dfs(u, v);
  }
}
inline int geth(int v, int k) {
  for (int i = 0; i < M; i++)
    if (k & (1 << i)) v = par[i][v];
  return v;
}
inline int lca(int u, int v) {
  if (d[u] > d[v]) swap(u, v);
  v = geth(v, d[v] - d[u]);
  if (u == v) return v;
  for (int i = M; i--;)
    if (par[i][v] != par[i][u]) u = par[i][u], v = par[i][v];
  return par[0][v];
}
inline int dis(int u, int v) { return d[u] + d[v] - 2 * d[lca(u, v)]; }
int main() {
  ios::sync_with_stdio(0), cin.tie(0);
  int n;
  cin >> n;
  for (int i = 1; i < n; i++) {
    int p;
    cin >> p;
    p--;
    adj[i].push_back(p);
    adj[p].push_back(i);
  }
  dfs(0, 0);
  int u = 0, v = 0, di = 0;
  for (int i = 1; i < n; i++) {
    int p = par[0][i];
    int d1 = dis(i, u);
    int d2 = dis(i, v);
    if (d1 > di) {
      v = u;
      u = i;
    } else if (d2 > di) {
      u = v;
      v = i;
    }
    di = dis(u, v);
    cout << di << " ";
  }
  cout << endl;
}
