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 N = 2e5 + 1;
int n, id = 0, p[N], h[N], l[N * 2], r[N * 2], euler[N * 2], LOG[N * 2];
vector<int> adj[N];
struct node {
  int h, v;
  bool operator<(const node &o) const { return h < o.h; }
} mn[19][N * 2];
void dfs(int v, int p) {
  euler[++id] = v;
  for (auto u : adj[v]) {
    if (h[u] == 0) {
      h[u] = h[v] + 1;
      dfs(u, v);
      euler[++id] = v;
    }
  }
}
int get_min(int l, int r) {
  int k = LOG[r - l + 1];
  return min(mn[k][l], mn[k][r - (1 << k) + 1]).v;
}
int lca(int u, int v) {
  if (u == v) {
    return u;
  }
  return get_min(min(l[u], l[v]), max(r[u], r[v]));
}
int simple_path(int u, int v) { return h[u] + h[v] - 2 * h[lca(u, v)]; }
int main() {
  cin.tie(0)->sync_with_stdio(0);
  cin >> n;
  for (int i = 2; i <= n; i++) {
    cin >> p[i];
    adj[p[i]].push_back(i);
  }
  for (int i = 1; i < N * 2; ++i) {
    if (i & 1) {
      LOG[i] = LOG[i - 1];
    } else {
      LOG[i] = LOG[i / 2] + 1;
    }
  }
  h[1] = 1;
  h[0] = -1;
  dfs(1, 0);
  for (int j = 0; j <= 18; ++j) {
    for (int i = 1; i <= id; ++i) {
      mn[j][i].h = INT_MAX;
    }
  }
  memset(l, 0x3f, sizeof l);
  for (int i = 1; i <= id; ++i) {
    int w = euler[i];
    l[w] = min(l[w], i);
    r[w] = max(r[w], i);
    mn[0][i] = {h[w], w};
  }
  for (int j = 1; j <= 18; ++j) {
    for (int i = 1; i <= id - (1 << j) + 1; ++i) {
      mn[j][i] = min(mn[j - 1][i], mn[j - 1][i + (1 << (j - 1))]);
    }
  }
  int u = 1, v = 1, ans = 0;
  for (int w = 2; w <= n; w++) {
    int newu = u, newv = v;
    if (ans < simple_path(u, w)) {
      ans = simple_path(u, w);
      newu = u;
      newv = w;
    }
    if (ans < simple_path(v, w)) {
      ans = simple_path(v, w);
      newu = v;
      newv = w;
    }
    u = newu;
    v = newv;
    cout << ans << ' ';
  }
}
