#include "roads.h"
#include <cmath>
#include <functional>
#include <fstream>
#include <iostream>
#include <vector>
#include <algorithm>
#include <string>
#include <set>
#include <map>
#include <list>
#include <time.h>
#include <math.h>
#include <random>
#include <deque>
#include <queue>
#include <unordered_map>
#include <unordered_set>
#include <iomanip>
#include <cassert>
#include <bitset>
#include <sstream>
#include <chrono>
#include <cstring>
#include <numeric>
using namespace std;
typedef long long ll;
#define all(x) x.begin(), x.end()
#define f first
#define s second
#define pb push_back
#define pii pair<int, int>
#define ve vector
#define vei vector<int>
#define vell vector<ll>
#define sz(x) ((int)x.size())
#define each(it, x) for (auto &it : x)
#define debug(x) cout << #x << " = " << x << "\n";
#define FU function
#define VO void
vell minimum_closure_costs(int N, vei U, vei V, vei W) {
int n = N;
assert(sz(U) == n - 1);
assert(sz(V) == n - 1);
assert(sz(W) == n - 1);
#ifdef ONPC
if (0) {
debug(n);
each(it, U) cout << it << " "; cout << "\n";
each(it, V) cout << it << " "; cout << "\n";
each(it, W) cout << it << " "; cout << "\n";
}
#endif
ve<ve<pii>> g(n);
for (int i = 0; i < n - 1; i++)
g[U[i]].pb({ V[i], W[i] }),
g[V[i]].pb({ U[i], W[i] });
vei ord, deg(n, 0), pap(n, -1), upp(n, 0);
for (int i = 0; i < n; i++) deg[i] = sz(g[i]);
vector<bool> vis(n, 0);
ord.push_back(0);
vis[0] = 1;
for (int i = 0; i < (int)ord.size(); i++)
{
int a = ord[i];
ve<pii> kids;
each(it, g[a])
{
int b = it.f, c = it.s;
if (vis[b]) continue;
pap[b] = a;
upp[b] = c;
vis[b] = 1;
ord.pb(b);
kids.pb({ b, c });
}
g[a] = kids;
}
assert((int)ord.size() == n);
reverse(all(ord));
vei loc(n);
for (int i = 0; i < n; i++) loc[ord[i]] = i;
for (int i = 0; i < n; i++)
sort(g[i].begin(), g[i].end(), [&](pair<int, int> i, pair<int, int> j) {return deg[i.first] > deg[j.first]; });
vector<vector<int>> devine(n);
for (int i = 0; i < n; i++)
{
devine[deg[i]].pb(i);
}
vell sol(n, 0);
vei verts;
vell dp0(n, 0), dp1(n, 0);
ll smsz = 0;
vector<ll> sui(n, 0);
vector<multiset<int>> gustav1(n), gustav2(n);
vector<ll> s1(n, 0);
for (int a = 0; a < n; a++) each(it, g[a]) sui[a] += it.second, gustav2[a].insert(-it.second);
auto mv12 = [&](int a)
{
assert(!gustav1[a].empty());
auto it = gustav1[a].end();
it--;
int x = *it;
gustav1[a].erase(it);
gustav2[a].insert(x);
s1[a] -= x;
};
auto mv21 = [&](int a)
{
assert(!gustav2[a].empty());
auto it = gustav2[a].begin();
int x = *it;
gustav2[a].erase(it);
gustav1[a].insert(x);
s1[a] += x;
};
for (int k = n - 1; k >= 0; k--)
{
each(x, devine[k])
{
verts.pb(x);
if (pap[x] != -1)
{
if (gustav1[pap[x]].find(-upp[x]) != gustav1[pap[x]].end())
{
s1[pap[x]] -= (-upp[x]);
gustav1[pap[x]].erase(gustav1[pap[x]].find(-upp[x]));
}
else
if (gustav2[pap[x]].find(-upp[x]) != gustav2[pap[x]].end())
{
gustav2[pap[x]].erase(gustav2[pap[x]].find(-upp[x]));
}
else assert(0);
sui[pap[x]] -= upp[x];
}
}
smsz += sz(verts);
sort(all(verts), [&](int a, int b) {return loc[a] < loc[b]; });
// 0 => are gradul <= k + 0
// 1 => are gradul <= k + 1
ll print = 0;
each(a, verts)
{
assert(deg[a] >= k);
each(rem, vei({ k - 1, k }))
{
// sterg muchia => dp1 + cost
// pastrez muchia => dp0
int total = sz(g[a]);
int pastrez = min(total, rem);
pastrez = max(pastrez, 0);
ll su = sui[a];
each(it, g[a]) if (deg[it.first] >= k) su += dp1[it.first] + it.second; else break;
vector<ll> difs;
each(it, g[a]) if (deg[it.first] >= k) difs.pb(min(0LL, dp0[it.first] - dp1[it.first] - it.second)); else break;
auto iter = gustav1[a].begin();
sort(difs.begin(), difs.end());
int po = 0;
ll best = (ll)1e18;
ll cur = 0;
assert((int)difs.size() + (int)gustav1[a].size() + (int)gustav2[a].size() >= pastrez);
for (int take = 0; take <= pastrez; take++)
{
if (take > (int)difs.size()) break;
if (take)
{
cur += difs[take - 1];
}
int from_gustav = pastrez - take;
if (from_gustav <= (int)gustav1[a].size() + (int)gustav2[a].size())
{
while ((int)gustav1[a].size() < from_gustav) mv21(a);
while ((int)gustav1[a].size() > from_gustav) mv12(a);
assert((int)gustav1[a].size() == from_gustav);
best = min(best, cur + s1[a]);
}
}
su += best;
if (rem == k - 1) dp0[a] = su; else dp1[a] = su;
}
if (a == 0)
{
print += dp1[a];
}
else
{
if (deg[pap[a]] < k)
{
print += min(dp0[a], dp1[a] + upp[a]);
}
}
assert(dp0[a] >= dp1[a]);
}
sol[k] = print;
}
assert(smsz <= 3 * n);
return sol;
}
