/****ThanhCodeDao*****/
/*******Azazel*******/
#include<bits/stdc++.h>
using namespace std;
#define F first
#define S second
#define pb push_back
#define checkbit(mask,i) ((mask>>i)&1)
#define bit(i) (1<<i)
#define MLog 17
typedef long long ll;
const ll maxN = 1e5+3, modd = 1e9+7;
struct Point {
ll x,y;
};
ll cross(Point p,Point q,Point r) { // vi tri r voi duong pq >0 la ben trai, =0 la thang hang
ll val = (q.x-p.x)*(r.y-q.y) - (q.y-p.y)*(r.x-q.x);
if(val < 0) return -1;
if(val > 0) return 1;
return 0;
}
ll dot(Point vecA,Point vecB) { // tra ve >0 la nhon, <0 la tu, =0 la vuong, 2 vecto phai chung goc
ll val = vecA.x*vecB.x + vecA.y*vecB.y;
if(val < 0) return -1;
if(val > 0) return 1;
return 0;
}
double triarea(Point p,Point q,Point r) { // cross(pq * pr) / 2
double cross = (q.x-p.x)*(r.y-q.y) - (q.y-p.y)*(r.x-q.x);
return fabs(cross)/2.0;
}
int n,q,a[maxN],f[maxN],h[maxN],cv[maxN];
vector<int> adj[maxN];
int pa[MLog+1][maxN];
int root = 1;
int st[maxN],fn[maxN],tour[2*maxN],cnt = 0;
priority_queue<pair<int,int>,vector<pair<int,int>>, greater<pair<int,int>>> pq;
void pre_dfs(int u,int par) {
st[u] = ++cnt;
tour[cnt] = u;
f[u] = a[u];
h[u] = h[par] + 1;
for(int v : adj[u]) {
if(v==par) continue;
pre_dfs(v,u);
f[u] = min(f[u],f[v]);
}
fn[u] = ++cnt;
tour[cnt] = u;
}
struct Fenwick {
vector<int> fen;
Fenwick(int n) {
fen.assign(n+3,0);
}
void up(int x,int delta) {
for(; x<=cnt; x+=x&-x) {
fen[x] += delta;
}
}
int get(int x) {
int res = 0;
for(; x>=1; x-=x&-x) {
res += fen[x];
}
return res;
}
int getlr(int l,int r) {
return get(r) - get(l-1);
}
};
bool cmp(int i,int j) {
return f[i] < f[j];
}
int times[maxN],curtime = 0;
void dfs(int u,int par) {
sort(adj[u].begin(),adj[u].end(),cmp);
for(int v : adj[u]) {
if(v==par) continue;
dfs(v,u);
}
times[u] = ++curtime;
pq.push({curtime,u});
}
int kpar(int u,int k) {
for(int i = MLog; i>=0; i--) {
if(k>=bit(i)) {
u = pa[i][u];
k -= bit(i);
}
}
return u;
}
void solve() {
cin >> n >> q;
for(int i = 1; i<=n; i++) cin >> a[i];
for(int i = 1; i<=n; i++) {
cin >> pa[0][i];
if(pa[0][i] == 0) {
root = i;
continue;
}
adj[i].pb(pa[0][i]);
adj[pa[0][i]].pb(i);
}
pre_dfs(root,0); // eulertour, mang f
for(int i = 1; i<=MLog; i++) {
for(int j = 1; j<=n; j++) {
pa[i][j] = pa[i-1][pa[i-1][j]];
}
}
dfs(root,0);
Fenwick fen(cnt);
while(q--) {
int type,u;
cin >> type >> u;
int res = 0;
if(type == 1) {
while(u) {
int v = pq.top().S;
pq.pop();
--u;
fen.up(st[v],1);
fen.up(fn[v],-1);
res = v;
}
} else {
res = fen.getlr(st[root],st[u])-1;
// tim cha cao nhat sao cho no get(1,cha) >= 1
int ans = kpar(u,res);
fen.up(st[ans],-1); // day viec cho u bang cach xoa ans
fen.up(fn[ans],+1);
pq.push({times[ans],ans});
}
cout << res << '\n';
}
}
int main() {
ios_base::sync_with_stdio(false);
cin.tie(0), cout.tie(0);
//freopen("inp.txt","r",stdin);
//freopen("out.txt","w",stdout);
solve();
return 0;
}
