#include<iostream>
#include<cstdio>
#include<vector>
#include<algorithm>
#include<cstdlib>
#include<random>
#define N 200000
#define M 4000000
using namespace std;
mt19937 RAND(random_device{}());
const long long inf=(long long)(1e18);
int read()
{
	char c=0;
	int sum=0;
	while (c<'0'||c>'9') c=getchar();
	while ('0'<=c&&c<='9') sum=sum*10+c-'0',c=getchar();
	return sum;
}
int n,q,dfn[N+1],leng,fa[N+1],rev[N+1],bt[N+1],a[N+1],b[N+1],sz[N+1],p[N+1],hs[N+1],top[N+1],rt[N+1],delta[N+1],delta2[N+1];
long long F[N+1],dp[N+1],DP[N+1],DP2[N+1];
vector<int>E[N+1];
bool cmp(int x,int y)
{
	return dp[x]>dp[y];
}
void dfs(int x)
{
	sz[x]=1;
	for (int i=0;i<E[x].size();++i)
	{
		fa[E[x][i]]=x,dfs(E[x][i]),sz[x]+=sz[E[x][i]];
		if (sz[E[x][i]]>sz[hs[x]]) hs[x]=E[x][i];
	}
	return;
}
void dfs2(int x)
{
	dfn[x]=++leng,rev[dfn[x]]=x;
	if (hs[x]) top[hs[x]]=top[x],dfs2(hs[x]),bt[x]=bt[hs[x]];
	else bt[x]=x;
	for (int i=0;i<E[x].size();++i)
		if (E[x][i]!=hs[x])
			top[E[x][i]]=E[x][i],dfs2(E[x][i]);
	return;
}
void dfs3(int x)
{
	dp[x]=F[x];
	for (int i=0;i<E[x].size();++i) dfs3(E[x][i]);
	for (int i=0;i<E[x].size();++i) p[i]=E[x][i];
	sort(p,p+E[x].size(),cmp);
	for (int i=0;i<E[x].size();++i)
		if (dp[x]<=dp[p[i]])
			dp[x]-=b[p[i]];
	return;
}
struct reads
{
	int num;
	long long data;
	bool operator < (const reads &t)const
	{
		return data!=t.data?data>t.data:num>t.num;	
	}
	bool operator > (const reads &t)const
	{
		return data!=t.data?data<t.data:num<t.num;	
	}
	bool operator <= (const reads &t)const
	{
		return data!=t.data?data>t.data:num>=t.num;	
	}
};
struct rds
{
	long long a,b;
};
rds operator + (rds x,rds y)
{
	return (rds){x.a+y.a,min(x.a+y.b,x.b)};
}
struct fhq_treap
{
	struct node
	{
		int ls,rs,rd;
		reads data;
		rds nw,res;
	};
	node tree[M+1];
	int length;
	int new_node(reads x)
	{
		++length,tree[length]=(node){0,0,(int)(RAND()%inf),x,(rds){b[x.num],x.data},(rds){b[x.num],x.data}};
		return length;
	}
	void push_up(int k)
	{
		tree[k].res=tree[tree[k].ls].res+tree[k].nw+tree[tree[k].rs].res;
		return;
	}
	int merge(int x,int y)
	{
		if (!x||!y) return x^y;
		if (tree[x].rd<tree[y].rd)
		{
			tree[x].rs=merge(tree[x].rs,y),push_up(x);
			return x;
		}
		else
		{
			tree[y].ls=merge(x,tree[y].ls),push_up(y);
			return y;
		}
	}
	void split(int k,reads d,int &x,int &y)
	{
		if (!k)
		{
			x=y=0;
			return;
		}
		if (tree[k].data<=d) split(tree[k].rs,d,tree[k].rs,y),x=k;
		else split(tree[k].ls,d,x,tree[k].ls),y=k;
		push_up(k);
		return;
	}
	void split2(int k,reads d,int &x,int &y)
	{
		if (!k)
		{
			x=y=0;
			return;
		}
		if (tree[k].data<d) split2(tree[k].rs,d,tree[k].rs,y),x=k;
		else split2(tree[k].ls,d,x,tree[k].ls),y=k;
		push_up(k);
		return;
	}
	int insert(int k,reads d)
	{
		int A,B;
		split(k,d,A,B);
		return merge(merge(A,new_node(d)),B);
	}
	int del(int k,reads d)
	{
		int tmp,A,B,C;
		split(k,d,tmp,C),split2(tmp,d,A,B);
		return merge(A,C);
	}
	long long calc(int k,long long x,rds d)
	{
		if (!k) return 0;
		if (x<=(d+tree[tree[k].ls].res+tree[k].nw).b) return tree[tree[k].ls].res.a+tree[k].nw.a+calc(tree[k].rs,x,d+tree[tree[k].ls].res+tree[k].nw);
		else if (x<=(d+tree[tree[k].ls].res).b) return tree[tree[k].ls].res.a;
		else return calc(tree[k].ls,x,d);
	}
};
fhq_treap T;
struct dst
{
	int p[2];
};
dst cs;
dst operator * (dst a,dst b)
{
	for (int i=0;i<=1;++i) cs.p[i]=b.p[a.p[i]];
	return cs;
}
struct seg
{
	struct node
	{
		int l,r;
		dst data;
	};
	node tree[(N<<2)+1];
	void push_up(int k)
	{
		tree[k].data=tree[k<<1|1].data*tree[k<<1].data;
		return;
	}
	void build(int k,int l,int r)
	{
		tree[k].l=l,tree[k].r=r;
		if (l==r)
		{
			tree[k].data.p[0]=delta[rev[l]],tree[k].data.p[1]=delta2[rev[l]];
			return;
		}
		int mid=(l+r)>>1;
		build(k<<1,l,mid),build(k<<1|1,mid+1,r),push_up(k);
		return;	
	}
	void add(int k,int x,dst d)
	{
		if (tree[k].l==tree[k].r)
		{
			tree[k].data=d;
			return;
		}
		int mid=(tree[k].l+tree[k].r)>>1;
		if (x<=mid) add(k<<1,x,d);
		else add(k<<1|1,x,d);
		push_up(k);
		return;
	}
	dst query(int k,int l,int r)
	{
		if (tree[k].l==l&&tree[k].r==r) return tree[k].data;
		int mid=(tree[k].l+tree[k].r)>>1;
		if (r<=mid) return query(k<<1,l,r);
		else if (l>=mid+1) return query(k<<1|1,l,r);
		else return query(k<<1|1,mid+1,r)*query(k<<1,l,mid);
	}
};
seg T2;
long long calc(int x)
{
	if (bt[x]==x) return a[x];
	int d=T2.query(1,dfn[x]+1,dfn[bt[x]]).p[0];
	if (!d) return DP[x];
	else return DP2[x];
}
void recover(int x,int y)
{
	if (hs[x]==y) return;
	rt[x]=T.del(rt[x],(reads){y,dp[y]}),dp[y]=calc(y),rt[x]=T.insert(rt[x],(reads){y,dp[y]});
	return;
}
void rebuild(int x)
{
	DP[x]=F[x]-T.calc(rt[x],F[x],(rds){0,inf}),DP2[x]=F[x]-b[hs[x]]-T.calc(rt[x],F[x]-b[hs[x]],(rds){0,inf});
	if (hs[x]) delta[hs[x]]=DP[hs[x]]>=DP[x],delta2[hs[x]]=DP2[hs[x]]>=DP[x],T2.add(1,dfn[hs[x]],(dst){delta[hs[x]],delta2[hs[x]]});
	if (fa[x]) delta[x]=DP[x]>=DP[fa[x]],delta2[x]=DP2[x]>=DP[fa[x]],T2.add(1,dfn[x],(dst){delta[x],delta2[x]});
	return;
}
void change(int p,int x,int y)
{
	F[p]+=x-a[p];
	if (fa[p]) F[fa[p]]+=y-b[p];
	a[p]=x,b[p]=y,rebuild(p);
	if (fa[p])
	{
		while (fa[p]) recover(fa[p],p),rebuild(fa[p]),p=top[fa[p]];
	}
	return;
}
int main()
{
	int op,p,x,y,c,d;
	n=read(),q=read(),T.tree[0].res=(rds){0,inf};
	for (int i=2;i<=n;++i) x=read(),E[x].push_back(i);
	for (int i=1;i<=n;++i) a[i]=read();
	for (int i=1;i<=n;++i) b[i]=read();
	for (int i=1;i<=n;++i)
	{
		F[i]=a[i];
		for (int j=0;j<E[i].size();++j) F[i]+=b[E[i][j]];
	}
	dfs(1),top[1]=1,dfs2(1),dfs3(1);
	for (int i=1;i<=n;++i)
		for (int j=0;j<E[i].size();++j)
			if (E[i][j]!=hs[i])
				rt[i]=T.insert(rt[i],(reads){E[i][j],dp[E[i][j]]});
	for (int i=1;i<=n;++i) DP[i]=F[i]-T.calc(rt[i],F[i],(rds){0,inf}),DP2[i]=F[i]-b[hs[i]]-T.calc(rt[i],F[i]-b[hs[i]],(rds){0,inf});
	for (int i=1;i<=n;++i)
		if (hs[i])
			delta[hs[i]]=DP[hs[i]]>=DP[i],delta2[hs[i]]=DP2[hs[i]]>=DP[i];
	T2.build(1,1,n);
	while (q--)
	{
		op=read();
		if (op==1) p=read(),x=read(),y=read(),change(p,x,y);
		else c=read(),d=read(),printf("%d\n",(reads){c,calc(c)}>(reads){d,calc(d)});
	}
	return 0;
}