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;
int n;
vector<int> G[200001];
int root = 0;
int parent[21][200001];
int depth[200001];
void dfs(int v, int p, int d) {
  parent[0][v] = p;
  depth[v] = d;
  for (int i = 0; i < G[v].size(); i++) {
    if (G[v][i] != p) {
      dfs(G[v][i], v, d + 1);
    }
  }
}
void init() {
  dfs(root, -1, 0);
  for (int k = 0; k + 1 < 20; k++) {
    for (int v = 0; v < n; v++) {
      if (parent[k][v] < 0) {
        parent[k + 1][v] = -1;
      } else {
        parent[k + 1][v] = parent[k][parent[k][v]];
      }
    }
  }
}
int lca(int u, int v) {
  if (depth[u] > depth[v]) {
    swap(u, v);
  }
  int res = 0;
  for (int k = 0; k < 20; k++) {
    if (((depth[v] - depth[u]) >> k) & 1) {
      res += 1 << k;
      v = parent[k][v];
    }
  }
  if (u == v) return res;
  for (int k = 19; k >= 0; k--) {
    if (parent[k][u] != parent[k][v]) {
      res += (1 << (k + 1));
      u = parent[k][u];
      v = parent[k][v];
    }
  }
  return res + 2;
}
int res[200001];
int main(void) {
  scanf("%d", &n);
  for (int i = 1; i <= n - 1; i++) {
    int a;
    scanf("%d", &a);
    a--;
    G[i].push_back(a);
    G[a].push_back(i);
  }
  init();
  int df = 0, dt = 1;
  res[0] = 1;
  for (int i = 2; i < n; i++) {
    if (lca(df, dt) < lca(i, df)) {
      dt = i;
    }
    if (lca(df, dt) < lca(i, dt)) {
      df = i;
    }
    res[i - 1] = lca(df, dt);
  }
  for (int i = 0; i < n - 1; i++) {
    printf("%d%c", res[i], i + 1 == n - 1 ? '\n' : ' ');
  }
  return 0;
}
