/*************************************
* author: marvinthang *
* created: 21.01.2024 19:34:55 *
*************************************/
#include "beechtree.h"
#include <bits/stdc++.h>
using namespace std;
#define fi first
#define se second
#define left ___left
#define right ___right
#define TIME (1.0 * clock() / CLOCKS_PER_SEC)
#define MASK(i) (1LL << (i))
#define BIT(x, i) ((x) >> (i) & 1)
#define __builtin_popcount __builtin_popcountll
#define ALL(v) (v).begin(), (v).end()
#define REP(i, n) for (int i = 0, _n = (n); i < _n; ++i)
#define REPD(i, n) for (int i = (n); i-- > 0; )
#define FOR(i, a, b) for (int i = (a), _b = (b); i < _b; ++i)
#define FORD(i, b, a) for (int i = (b), _a = (a); --i >= _a; )
#define FORE(i, a, b) for (int i = (a), _b = (b); i <= _b; ++i)
#define FORDE(i, b, a) for (int i = (b), _a = (a); i >= _a; --i)
#define scan_op(...) istream & operator >> (istream &in, __VA_ARGS__ &u)
#define print_op(...) ostream & operator << (ostream &out, const __VA_ARGS__ &u)
#ifdef LOCAL
#include "debug.h"
#else
#define file(name) if (fopen(name".inp", "r")) { freopen(name".inp", "r", stdin); freopen(name".out", "w", stdout); }
#define DB(...) 23
#define db(...) 23
#define debug(...) 23
#endif
template <class U, class V> scan_op(pair <U, V>) { return in >> u.first >> u.second; }
template <class T> scan_op(vector <T>) { for (size_t i = 0; i < u.size(); ++i) in >> u[i]; return in; }
template <class U, class V> print_op(pair <U, V>) { return out << '(' << u.first << ", " << u.second << ')'; }
template <size_t i, class T> ostream & print_tuple_utils(ostream &out, const T &tup) { if constexpr(i == tuple_size<T>::value) return out << ")"; else return print_tuple_utils<i + 1, T>(out << (i ? ", " : "(") << get<i>(tup), tup); }
template <class ...U> print_op(tuple<U...>) { return print_tuple_utils<0, tuple<U...>>(out, u); }
template <class Con, class = decltype(begin(declval<Con>()))> typename enable_if <!is_same<Con, string>::value, ostream&>::type operator << (ostream &out, const Con &con) { out << '{'; for (__typeof(con.begin()) it = con.begin(); it != con.end(); ++it) out << (it == con.begin() ? "" : ", ") << *it; return out << '}'; }
// end of template
vector <int> beechtree(int N, int M, vector <int> P, vector <int> C) {
vector <vector <int>> adj(N);
vector <int> sz(N, 1), res(N, 1), heavy(N, -1);
vector <bool> is_head(N, true);
REPD(u, N) {
--C[u];
if (heavy[u] != -1) is_head[heavy[u]] = false;
if (u) {
sz[P[u]] += sz[u];
adj[P[u]].push_back(u);
if (heavy[P[u]] == -1 || sz[heavy[P[u]]] < sz[u]) heavy[P[u]] = u;
}
}
vector <pair <int, int>> val;
vector <int> cnt(N + 1);
REP(u, N) if (is_head[u]) {
val.emplace_back(u, u);
++cnt[sz[u]];
for (int i = (int) val.size() - 1; i < (int) val.size(); ++i) {
for (int &v: adj[val[i].fi]) {
++cnt[sz[v]];
val.emplace_back(v, u);
}
}
}
partial_sum(ALL(cnt), cnt.begin());
vector <pair <int, int>> sorted_val(cnt.back());
vector <vector <int>> order(N);
for (auto &v: val) sorted_val[--cnt[sz[v.fi]]] = v;
for (auto &[v, u]: sorted_val) order[u].push_back(v);
vector <int> tin(N), tout(N);
function<void(int)> dfs1 = [&] (int u) {
static int t = 0;
tin[u] = t++;
for (int &v: adj[u]) dfs1(v);
tout[u] = t;
};
dfs1(0);
vector <int> color_sz(M, 0);
REPD(u, N) {
if (res[u]) {
for (int &v: adj[u]) {
if (color_sz[C[v]]) {
res[u] = 0;
break;
}
color_sz[C[v]] = 1;
}
for (int &v: adj[u]) color_sz[C[v]] = 0;
}
if (u && !res[u]) res[P[u]] = 0;
}
function<void(int)> dfs2 = [&] (int u) {
auto check_subtree = [&] (int v) {
int last = -1;
for (int &w: order[u]) if (tin[v] <= tin[w] && tin[w] < tout[v]) {
int cnt = (int) adj[last].size();
for (int &v: adj[w]) {
if (color_sz[C[v]] > sz[v]) {
if (last != -1) for (int &v: adj[last]) color_sz[C[v]] = 0;
return true;
}
cnt -= !!color_sz[C[v]];
}
if (last != -1 && cnt) {
for (int &v: adj[last]) color_sz[C[v]] = 0;
return true;
}
for (int &v: adj[w]) color_sz[C[v]] = sz[v];
last = w;
}
for (int &v: adj[last]) color_sz[C[v]] = 0;
return false;
};
vector <int> path;
for (int v = u; v != -1; v = heavy[v]) path.push_back(v);
int tl = 0;
while (tl < (int) path.size() && !res[path[tl]]) ++tl;
auto p = partition_point(tl + ALL(path), check_subtree);
if (p != path.end()) res[*p] = 1;
for (auto it = path.begin(); it != p; ++it) {
for (int &v: adj[*it]) if (v != heavy[*it]) dfs2(v);
res[*it] = 0;
}
};
dfs2(0);
FOR(i, 1, N) if (res[P[i]]) res[i] = 1;
return res;
}
