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;
struct Set {
  int pa;
  int sz;
  int a, b;
} pa[200010];
int findset(int x) { return pa[x].pa == x ? x : pa[x].pa = findset(pa[x].pa); }
int p[200010], depth[200010], fa[200010][25];
vector<int> G[200010];
void dfs(int u) {
  for (int i = 0; i < G[u].size(); i++) {
    int v = G[u][i];
    if (v == fa[u][0]) continue;
    fa[v][0] = u;
    depth[v] = depth[u] + 1;
    dfs(v);
  }
}
int lca(int u, int v) {
  if (depth[u] < depth[v]) swap(u, v);
  for (int i = 20; i >= 0; i--) {
    if (depth[fa[u][i]] >= depth[v]) {
      u = fa[u][i];
    }
  }
  if (u == v) return u;
  for (int i = 20; i >= 0; i--) {
    if (fa[u][i] != fa[v][i]) {
      u = fa[u][i];
      v = fa[v][i];
    }
  }
  return fa[u][0];
}
int dist(int u, int v) { return depth[u] + depth[v] - 2 * depth[lca(u, v)]; }
int main() {
  int n;
  scanf("%d", &n);
  for (int i = 2; i <= n; i++) {
    scanf("%d", &p[i]);
    G[i].push_back(p[i]);
    G[p[i]].push_back(i);
  }
  depth[1] = 1;
  dfs(1);
  for (int i = 1; i < 21; i++) {
    for (int j = 1; j <= n; j++) fa[j][i] = fa[fa[j][i - 1]][i - 1];
  }
  for (int i = 1; i <= n; i++) {
    pa[i].pa = i;
    pa[i].sz = 1;
    pa[i].a = i;
  }
  int best = 0;
  for (int i = 2; i <= n; i++) {
    int u = i, v = p[i];
    int x = findset(u), y = findset(v);
    vector<int> box;
    if (pa[x].sz == 1)
      box.push_back(pa[x].a);
    else {
      box.push_back(pa[x].a);
      box.push_back(pa[x].b);
    }
    if (pa[y].sz == 1)
      box.push_back(pa[y].a);
    else {
      box.push_back(pa[y].a);
      box.push_back(pa[y].b);
    }
    int ret = 0, node1, node2;
    for (int j = 0; j < box.size(); j++) {
      for (int i = 0; i < box.size(); i++)
        if (i != j) {
          if (dist(box[i], box[j]) > ret) {
            ret = dist(box[i], box[j]);
            node1 = box[i], node2 = box[j];
          }
        }
    }
    pa[y].pa = x;
    pa[x].sz += pa[y].sz;
    pa[x].a = node1, pa[x].b = node2;
    best = max(best, ret);
    if (i > 2) putchar(' ');
    printf("%d", best);
  }
  puts("");
  return 0;
}
