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 = 200005;
const int L = 18;
int n, x[N], par[L][N], tin[N], tout[N], timer, d[N];
vector<int> g[N];
void dfs(int v, int pr = 0) {
  tin[v] = timer++;
  par[0][v] = pr;
  for (int i = 1; i < L; ++i) {
    par[i][v] = par[i - 1][par[i - 1][v]];
  }
  for (auto& to : g[v]) {
    if (to != pr) {
      d[to] = d[v] + 1;
      dfs(to, v);
    }
  }
  tout[v] = timer++;
}
int line(int x, int y) { return tin[x] <= tin[y] && tout[x] >= tout[y]; }
int lca(int x, int y) {
  if (line(x, y)) return x;
  if (line(y, x)) return y;
  for (int i = L - 1; i >= 0; --i) {
    if (!line(par[i][x], y)) x = par[i][x];
  }
  return par[0][x];
}
int dist(int x, int y) { return d[x] + d[y] - 2 * d[lca(x, y)]; }
int main() {
  scanf("%d", &n);
  for (int i = 1; i < n; ++i) {
    scanf("%d", x + i);
    --x[i];
    g[x[i]].push_back(i);
    g[i].push_back(x[i]);
  }
  dfs(0);
  pair<int, int> best = {0, 0};
  for (int i = 1; i <= n - 1; ++i) {
    if (dist(best.first, i) > dist(best.first, best.second)) {
      best = {best.first, i};
    }
    if (dist(best.second, i) > dist(best.first, best.second)) {
      best = {best.second, i};
    }
    printf("%d ", dist(best.first, best.second));
  }
  printf("\n");
  return 0;
}
