思路

第一个可以倍增，第二个讨论在a到lca的路径上还是lca到b的路径上，
倍增即可

代码

#include <cstdio>
#include <algorithm>
#include <cstring>
using namespace std;
int jump[10010][16],sum[10010][16],fir[10010],nxt[10010*2],v[10010*2],w[10010*2],cnt,dep[10010],n;
void addedge(int ui,int vi,int wi){++cnt;v[cnt]=vi;w[cnt]=wi;nxt[cnt]=fir[ui];fir[ui]=cnt;
}
void dfs(int u,int f,int wx){dep[u]=dep[f]+1;jump[u][0]=f;sum[u][0]=wx;for(int i=1;i<15;i++)jump[u][i]=jump[jump[u][i-1]][i-1],sum[u][i]=sum[u][i-1]+sum[jump[u][i-1]][i-1];for(int i=fir[u];i;i=nxt[i]){if(v[i]==f)continue;dfs(v[i],u,w[i]);}
}
int lca(int x,int y){if(dep[x]<dep[y])swap(x,y);for(int i=15;i>=0;i--)if(dep[jump[x][i]]>=dep[y])x=jump[x][i];if(x==y)return x;for(int i=15;i>=0;i--)if(jump[x][i]!=jump[y][i])x=jump[x][i],y=jump[y][i];return jump[x][0];
}
int query_sum(int x,int y){int ans=0;if(dep[x]<dep[y])swap(x,y);for(int i=15;i>=0;i--)if(dep[jump[x][i]]>=dep[y]){ans+=sum[x][i];x=jump[x][i];}if(x==y)return ans;for(int i=15;i>=0;i--)if(jump[x][i]!=jump[y][i]){ans+=sum[x][i]+sum[y][i];x=jump[x][i],y=jump[y][i];}return ans+sum[x][0]+sum[y][0];
}
int kth_fa(int x,int k){for(int i=15;i>=0;i--){if((k>>i)&1)x=jump[x][i];}return x;
}
void init(void){memset(jump,0,sizeof(jump));memset(sum,0,sizeof(sum));memset(fir,0,sizeof(fir));memset(nxt,0,sizeof(nxt));memset(v,0,sizeof(v));memset(w,0,sizeof(w));cnt=0;memset(dep,0,sizeof(dep));
}
int main(){int T;scanf("%d",&T);while(T--){init();scanf("%d",&n);for(int i=1;i<n;i++){int a,b,c;scanf("%d %d %d",&a,&b,&c);addedge(a,b,c);addedge(b,a,c);}dfs(1,0,0);char opt[10];while(1){scanf("%s",opt);if(opt[1]=='O')break;if(opt[1]=='I'){int a,b;scanf("%d %d",&a,&b);printf("%d\n",query_sum(a,b));}else{int a,b,c;scanf("%d %d %d",&a,&b,&c);int LCA=lca(a,b);if(dep[a]-dep[LCA]+1>=c){printf("%d\n",kth_fa(a,c-1));}else{c-=dep[a]-dep[LCA];int t2=dep[b]-dep[LCA]+1;printf("%d\n",kth_fa(b,t2-c));}}}}return 0;
}