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 Z = (int)2e5 + 111;
const int inf = (int)1e9 + 111;
const long long llinf = (long long)1e18 + 5;
const int MOD = (int)1e9 + 7;
vector<int> g[Z];
int l, up[Z][25], d[Z], tin[Z], tout[Z], timer, x[Z];
void dfs(int v, int p, int depth) {
  d[v] = depth;
  tin[v] = ++timer;
  up[v][0] = p;
  for (int i = 1; i <= l; ++i) {
    up[v][i] = up[up[v][i - 1]][i - 1];
  }
  for (int to : g[v]) {
    if (to == p) continue;
    dfs(to, v, depth + 1);
  }
  tout[v] = ++timer;
}
bool is(int a, int b) { return tin[a] <= tin[b] && tout[a] >= tout[b]; }
int lca(int a, int b) {
  if (is(a, b)) return a;
  if (is(b, a)) return b;
  for (int i = l; i >= 0; --i) {
    if (!is(up[a][i], b)) a = up[a][i];
  }
  return up[a][0];
}
int dist(int a, int b) { return d[a] + d[b] - 2 * d[lca(a, b)]; }
int main() {
  srand(time(0));
  ios_base::sync_with_stdio(false);
  int n;
  cin >> n;
  for (int i = 1; i < n; ++i) {
    int x;
    cin >> x;
    g[x].push_back(i + 1);
  }
  l = 1;
  while ((1 << l) <= n) ++l;
  dfs(1, 1, 0);
  int v1 = 1, v2 = 1;
  for (int i = 1; i < n; ++i) {
    int v3 = i + 1;
    int len1 = dist(v1, v2);
    int len2 = dist(v1, v3);
    int len3 = dist(v2, v3);
    int mx = max(max(len1, len2), len3);
    cout << mx << ' ';
    if (len1 == mx) continue;
    if (len2 == mx) {
      v2 = v3;
    } else if (len3 == mx) {
      v1 = v3;
    }
  }
  return 0;
}
