大体思路和楼上各位差不多，提供一种$O(n \log n)$的懒人写法

#include<bits/stdc++.h>
#define int long long
using namespace std;
int beg[20005],nex[20005],to[20005],d[20005],w[20005],f[20005],ans[20005],n,e,ma;
priority_queue<pair<int,int> >q[20005];
void add(int x,int y,int z){
    to[++e]=y;
    nex[e]=beg[x];
    beg[x]=e;
    w[e]=z;
}
void dfs_anyroot(int x,int fa){
    q[x].push(make_pair(0,0));
    for(int i=beg[x];i;i=nex[i])
        if(fa!=to[i]){
            int y=to[i];
            d[y]=d[x]+w[i];
            dfs_anyroot(y,x);
            f[x]=max(f[x],f[y]+w[i]);
            q[x].push(make_pair(f[y]+w[i],y));
        }
}
void dfs(int x,int fa){
    for(int i=beg[x];i;i=nex[i])
        if(to[i]!=fa){
            int y=to[i],qwq,ff=0;
            if(q[x].top().second==y)qwq=q[x].top().first,ff=1,q[x].pop();
            ans[y]=max(f[y],q[x].top().first+w[i]);
            q[y].push(make_pair(q[x].top().first+w[i],x));
            if(ff)q[x].push(make_pair(qwq,y));
            dfs(y,x);
        }
    while(!q[x].empty())q[x].pop();
} 
signed main(){
    while(scanf("%lld",&n)!=EOF){
        memset(nex,0,sizeof(nex));
        memset(to,0,sizeof(to));
        memset(ans,0,sizeof(ans));
        memset(f,0,sizeof(f));
        memset(beg,0,sizeof(beg));
        memset(d,0,sizeof(d));
        memset(w,0,sizeof(w));
        e=0;
        ma=-1;
        int x,y,z;
        for(int i=2;i<=n;i++){
            scanf("%lld%lld",&x,&y);
            add(i,x,y);
            add(x,i,y);
        }
        dfs_anyroot(1,0);
        ans[1]=f[1];
        dfs(1,0);
        for(int i=1;i<=n;i++)
            printf("%lld\n",ans[i]);
    }
    return 0;
}