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>
const int MAXN = 210001;
const int MAXM = 410001;
struct Edge {
  int node, next;
} e[MAXM];
int n, m, t, h[MAXN], f[MAXN][21], d[MAXN];
void addEdge(int x, int y) {
  t++;
  e[t] = (Edge){y, h[x]};
  h[x] = t;
}
void dfs(int x) {
  for (int i = h[x]; i; i = e[i].next) {
    d[e[i].node] = d[x] + 1;
    dfs(e[i].node);
  }
}
int getanc(int x, int t) {
  for (int i = 0; i <= 20; i++)
    if ((t >> i) & 1) x = f[x][i];
  return x;
}
int getlca(int x, int y) {
  if (d[x] > d[y]) x = getanc(x, d[x] - d[y]);
  if (d[x] < d[y]) y = getanc(y, d[y] - d[x]);
  if (x == y) return x;
  for (int i = 20; i >= 0; i--)
    if (f[x][i] != f[y][i]) {
      x = f[x][i];
      y = f[y][i];
    }
  return f[x][0];
}
int getdist(int x, int y) { return d[x] + d[y] - 2 * d[getlca(x, y)]; }
int main() {
  scanf("%d", &n);
  for (int i = 2; i <= n; i++) {
    scanf("%d", &f[i][0]);
    addEdge(f[i][0], i);
  }
  dfs(1);
  for (int j = 1; j <= 20; j++)
    for (int i = 1; i <= n; i++) f[i][j] = f[f[i][j - 1]][j - 1];
  std::pair<int, std::pair<int, int> > answer =
      std::make_pair(0, std::make_pair(1, 1));
  for (int i = 2; i <= n; i++) {
    std::pair<int, std::pair<int, int> > tmp1 =
        std::make_pair(getdist(i, answer.second.first),
                       std::make_pair(i, answer.second.first));
    std::pair<int, std::pair<int, int> > tmp2 =
        std::make_pair(getdist(i, answer.second.second),
                       std::make_pair(i, answer.second.second));
    answer = std::max(answer, tmp1);
    answer = std::max(answer, tmp2);
    printf("%d%c", answer.first, " \n"[i == n]);
  }
  return 0;
}
