#include<bits/stdc++.h>
#define LL long long
#define LLL __int128
#define uint unsigned
#define ldb long double
#define uLL unsigned long long
using namespace std;
const int N=2e5+5;
const LL INF=1e16;
int n,q,tim;
vector<int>G[N];
int a[N],b[N];LL sb[N],Q[N];
int f[N],top[N],sz[N],son[N],dL[N],dR[N],rk[N],rt[N];
mt19937 rng(chrono::system_clock::now().time_since_epoch().count());
struct node{
    LL p,l,r;
    node(const LL&_p=0,const LL&_l=0,const LL&_r=0):
        p(_p),l(_l),r(_r){}
    inline LL operator()(const LL&x)const{
        return x<p?l:r;
    }
    inline node operator+(const node&o)const{
        return node(o.p,(o.l<p?l:r),(o.r<p?l:r));
    }
}A[N],S[N*4];
struct Node{
    LL v,a,mn,s;int pri;
    inline void init(LL _v,LL _a){
        v=_v,a=_a,mn=_v,s=_a,pri=rng();
    }
}T[N];
int Ls[N],Rs[N];
inline void pushup(int p){
    T[p].s=T[Ls[p]].s+T[p].a+T[Rs[p]].s;
    T[p].mn=min(min(T[Ls[p]].mn,T[Ls[p]].s+T[p].v),T[Ls[p]].s+T[p].a+T[Rs[p]].mn);
}
inline int merge(int x,int y){
    if(!x||!y)return x|y;
    if(T[x].pri>T[y].pri)return Rs[x]=merge(Rs[x],y),pushup(x),x;
    else return Ls[y]=merge(x,Ls[y]),pushup(y),y;
}
inline void split(int x,LL k,int r,int&p,int&q){
    if(!x)return p=0,q=0,void();
    if(T[x].v>k||(T[x].v==k&&x>r))p=x,split(Rs[x],k,r,Rs[p],q),pushup(p);
    else q=x,split(Ls[x],k,r,p,Ls[q]),pushup(q);
}
inline LL qry(int rt,LL x){
    if(!rt)return x;
    if(x>T[Ls[rt]].mn)return qry(Ls[rt],x);
    if(x>T[Ls[rt]].s+T[rt].v)return x-T[Ls[rt]].s;
    return qry(Rs[rt],x-T[Ls[rt]].s-T[rt].a);
}
inline void ins(int &rt,int x){
    int p,q;split(rt,T[x].v,x,p,q);
    rt=merge(p,merge(x,q));
}
inline void del(int &rt,int x){
    int p,q;split(rt,T[x].v,x,p,q),split(q,T[x].v,x-1,rt,q);
    rt=merge(p,q);
}
inline void build(int p,int l,int r){
    if(l==r)return S[p]=A[rk[l]],void();
    const int mid=(l+r)>>1,ls=p*2,rs=p*2+1;
    build(ls,l,mid),build(rs,mid+1,r);
    S[p]=S[ls]+S[rs];
}
inline void upd(int p,int l,int r,int x){
    if(l==r)return S[p]=A[rk[l]],void();
    const int mid=(l+r)>>1,ls=p*2,rs=p*2+1;
    x<=mid?upd(ls,l,mid,x):upd(rs,mid+1,r,x);
    S[p]=S[ls]+S[rs];
}
inline node qry(int p,int l,int r,int x){
    if(x<=l)return S[p];
    const int mid=(l+r)>>1,ls=p*2,rs=p*2+1;
    return x<=mid?qry(ls,l,mid,x)+qry(rs,mid+1,r,x):qry(rs,mid+1,r,x);
}
inline void dfs1(int u,int fa){
    sz[u]=1,son[u]=0,f[u]=fa;
    for(auto v:G[u])if(v!=fa)
        dfs1(v,u),sz[u]+=sz[v],sb[u]+=b[v],
        sz[son[u]]<sz[v]?son[u]=v:0;
    for(auto v:G[u])if(v!=son[u])
        T[v].init(Q[v],b[v]),ins(rt[u],v);
    if(son[u]){
        auto x=qry(rt[u],sb[u]+a[u]);
        auto y=qry(rt[u],sb[u]+a[u]-b[son[u]]);
        A[u]=node(x,x,y);
    }
    else A[u]=node(0,a[u],a[u]);
    Q[u]=A[u](Q[son[u]]);
}
inline void dfs2(int u,int fa,int tp){
    rk[dL[u]=++tim]=u,top[u]=tp;
    if(son[u])dfs2(son[u],u,tp);
    for(auto v:G[u])if(v!=fa&&v!=son[u])dfs2(v,u,v);
    dR[u]=tim;
}
inline void upd(int u){
    for(int i=u;;i=f[top[i]]){
        if(son[i]){
            auto x=qry(rt[i],sb[i]+a[i]);
            auto y=qry(rt[i],sb[i]+a[i]-b[son[i]]);
            A[i]=node(x,x,y);
        }
        else A[i]=node(0,a[i],a[i]);
        upd(1,1,n,dL[i]);
        if(!f[top[i]])break;
        del(rt[f[top[i]]],top[i]);
        T[top[i]].init(qry(1,1,n,dL[top[i]])(0),b[top[i]]);
        ins(rt[f[top[i]]],top[i]);
    }
}
signed main(){
    cin.tie(0)->sync_with_stdio(0);
    T[0].v=-INF,T[0].mn=INF;
    cin>>n>>q;
    for(int i=2;i<=n;++i)
        cin>>f[i],G[f[i]].emplace_back(i);
    for(int i=1;i<=n;++i)cin>>a[i];
    for(int i=1;i<=n;++i)cin>>b[i];
    dfs1(1,0),dfs2(1,0,1),build(1,1,n);
    for(int i=1;i<=q;++i){
        int o;cin>>o;
        if(o==1){
            int u,x,y;cin>>u>>x>>y;
            a[u]=x;
            upd(u);
            if(f[u]){
                sb[f[u]]+=y-b[u],b[u]=y;
                if(u!=son[f[u]]){
                    del(rt[f[u]],u);
                    T[u].init(qry(1,1,n,dL[u])(0),b[u]);
                    ins(rt[f[u]],u);
                }
                upd(f[u]);
            }
        }
        else{
            int u,v;cin>>u>>v;
            cout<<(make_pair(qry(1,1,n,dL[u])(0),u)<make_pair(qry(1,1,n,dL[v])(0),v))<<'\n';
        }
    }
    return 0;
}
/*
*/