#include<iostream>
#include<cstdio>
#include<algorithm>
#include<vector>
#define ll long long
using std::cin;using std::cout;
constexpr int N=100005,M=2000005;
constexpr ll inf=1e18;
int n,a[N],b[N],u,v,rt[M],ls[M],rs[M],cnt,c=N,d=-N,s[M],tot;
ll ans[N];
std::vector<int>edge[N];
struct Line{int k;ll b;}f[N];
inline ll calc(int x,int k){
	// cout<<"(calc)"<<x<<' '<<k<<'\n';
	return 1ll*f[k].k*(x+c-1)+f[k].b;
}
inline int cmp(ll a,ll b){if(a>b) return 1;if(a<b) return -1;return 0;}
inline int change(int l,int r,int k,int p){
	if(!p){s[p=++cnt]=k;return p;}
	// cout<<"(change)"<<l<<' '<<r<<' '<<k<<' '<<p<<'\n';
	int &v=s[p],mid=(l+r)/2,bmid=cmp(calc(mid,k),calc(mid,v));
	if(bmid==-1) std::swap(k,v);
	int bl=cmp(calc(l,k),calc(l,v)),br=cmp(calc(r,k),calc(r,v));
	if(bl==-1) ls[p]=change(l,mid,k,ls[p]);
	if(br==-1) rs[p]=change(mid+1,r,k,rs[p]);
	return p;
}
inline int merge(int a,int b,int l,int r){
	if(!a) return b;
	if(!b) return a;
	change(l,r,a,s[b]);
	int mid=(l+r)/2;
	ls[a]=merge(ls[a],ls[b],l,mid);
	rs[a]=merge(rs[a],rs[b],mid+1,r);
	return a;
}
// inline int update(int l,int r,int x,int y,int k,int p){
// 	if(!p) p=++cnt;
// 	if(x<=l&&r<=y) return change(l,r,k,p);
// 	int mid=(l+r)>>1;
// 	if(x<=mid) ls[p]=update(l,mid,x,y,k,ls[p]);
// 	if(mid<y) rs[p]=update(mid+1,r,x,y,k,rs[p]);
// 	return p;
// }
inline ll query(int l,int r,int x,int p){
	if(!p) return inf;
	int mid=(l+r)/2;
	ll now=calc(x,s[p]);
	// cout<<"(query)"<<l<<' '<<r<<' '<<x<<' '<<p<<'\n';
	// if(l==r) return now;
	if(x<=mid) return std::min(now,query(l,mid,x,ls[p]));
	return std::min(now,query(mid+1,r,x,rs[p]));
}
inline void down(int u,int fa){
	bool son=0;
	for(int i:edge[u]){
		if(i==fa) continue;
		down(i,u);son=1;
		rt[u]=merge(rt[u],rt[i],1,d);
	}
	if(son) ans[u]=query(1,d,a[u]+1-c,rt[u]);
	f[++tot]={b[u],ans[u]};
	// cout<<"f["<<tot<<"]:"<<b[u]<<' '<<ans[u]<<'\n';
	if(!rt[u]) rt[u]=++cnt;
	change(1,d,tot,rt[u]);
	// rt[u]=update(c,d,c,d,cnt,rt[u]);
}
signed main(){
//	freopen(".in","r",stdin);
//	freopen(".out","w",stdout);
	std::ios::sync_with_stdio(false);
	cin.tie(nullptr);cout.tie(nullptr);
	cin>>n;f[0].b=inf;
	for(int i=1;i<=n;++i){cin>>a[i];c=std::min(a[i],c);d=std::max(a[i],d);}
	d+=1-c;
	for(int i=1;i<=n;++i) cin>>b[i];
	for(int i=1;i<n;++i){cin>>u>>v;edge[u].push_back(v);edge[v].push_back(u);}
	down(1,0);for(int i=1;i<=n;++i) cout<<ans[i]<<' ';
	return 0;
}
// f[x]=f[y]+a[x]*b[y]=b[y]*a[x]+f[y]