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;
vector<vector<int>> t;
vector<vector<int>> f;
vector<int> tin, tout, d;
int timer = 0;
bool is_a(int u, int v) { return tin[u] <= tin[v] && tout[u] >= tout[v]; }
void dfs(int i = 0, int fth = 0, int dth = 0) {
  tin[i] = ++timer;
  f[0][i] = fth;
  d[i] = dth;
  for (int adj : t[i]) {
    if (adj == fth) continue;
    dfs(adj, i, dth + 1);
  }
  tout[i] = ++timer;
}
int lca(int u, int v) {
  if (is_a(u, v)) return u;
  if (is_a(v, u)) return v;
  for (int i = 19; i >= 0; --i)
    if (!is_a(f[i][u], v)) u = f[i][u];
  return f[0][u];
}
int main() {
  int n;
  cin >> n;
  t.resize(n);
  tin.resize(n);
  tout.resize(n);
  f.resize(20);
  d.resize(n);
  vector<int> ans;
  for (int i = 0; i < 20; ++i) f[i].resize(n);
  for (int i = 1; i <= n - 1; ++i) {
    int u;
    cin >> u;
    u--;
    t[i].push_back(u);
    t[u].push_back(i);
  }
  dfs();
  for (int i = 1; i < 20; ++i) {
    for (int j = 0; j < n; ++j) f[i][j] = f[i - 1][f[i - 1][j]];
  }
  int a = 0, b = 0, da = 0;
  for (int i = 1; i < n; ++i) {
    int anc1 = lca(a, i), d1 = d[a] + d[i] - 2 * d[anc1];
    int anc2 = lca(b, i), d2 = d[b] + d[i] - 2 * d[anc2];
    if (d1 > da && d1 > d2)
      ans.push_back(d1), b = i, da = d1;
    else if (d2 > da)
      ans.push_back(d2), a = i, da = d2;
    else
      ans.push_back(da);
  }
  for (int k : ans) cout << k << " ";
}
