Problem: 2063C - Remove Exactly Two
Language: c++17 

Solution 1 (runtime: 249 ms):
#include <bits/stdc++.h>
using namespace std;

#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>

#define ll long long int
#define ull unsigned long long int

#define M1 1000000007
#define M2 998244353
#define nl "\n"

template<class T> using ordered_set = __gnu_pbds::tree<T, __gnu_pbds::null_type, less<T>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>;
template<class T> using ordered_multiset = __gnu_pbds::tree<T, __gnu_pbds::null_type, less_equal<T>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>;

template<class T> using reverse_ordered_set = __gnu_pbds::tree<T, __gnu_pbds::null_type, greater<T>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>;
template<class T> using reverse_ordered_multiset = __gnu_pbds::tree<T, __gnu_pbds::null_type, greater_equal<T>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>;

// A.find_by_order(a)     Iterator to the first element of array A, where (Ai=a)
// A.order_of_key(a)      Number of elements type(less/greater) than a

ll binExp(ll x, ll n, ll m){ ll ans = 1; while (n) { if (n & 1) ans = ans * x % m; x = x * x % m; n >>= 1; } return ans;}
ll modAdd(ll a, ll b, ll m){ return (a % m + b % m) % m; }
ll modSub(ll a, ll b, ll m){ return ((a % m - b % m) + m) % m; }
ll modMul(ll a, ll b, ll m){ return (a % m * b % m) % m; }
ll modInverse(ll A, ll m){ return binExp(A, m - 2, m); }
ll modDiv(ll a, ll b, ll m) { return modMul(a % m, modInverse(b % m, m), m) % m ;}
ll fact(ll n, ll m){ if (n == 0) return 1; ll ans = 1; for (ll i = 2; i <= n; i++) ans = modMul(ans, i, m); return ans; }
ll fact(ll n){ if (n == 0) return 1; ll ans = 1; for (ll i = 2; i <= n; i++) ans *= i; return ans; }
string toBinary(ll n){ string b = ""; for (ll i = 59; i >= 0; i--) { ll k = n >> i; if (k & 1) b += '1'; else b += '0'; } return b; }
ll toDecimal(string &s) { ll ans = 0; vector<ll> pow(60, 1); for (int i = 1; i < 60; i++) pow[i] = pow[i-1] * 2; for (ll i = 59; i >= 0; i--) { if (s[i] == '1') ans += pow[59 - i]; } return ans; }
void computeFactors(ll n, vector<ll> &f){ for (ll i = 1; i <= sqrt(n); i++) { if(n % i == 0){ f.push_back(i); if(i != (n / i)) f.push_back(n / i); }}}

class DSU{
    vector<ll> size;
    vector<ll> par;

public:
    DSU(ll n){
        size.resize(n, 1);
        par.resize(n);
        for (int i = 0; i < n; i++)
            par[i] = i;
    }

    ll findUParent(ll x){
        if(x == par[x])
            return x;
        par[x] = findUParent(par[x]);
        return par[x];
    }

    void makeUnion(ll a, ll b){
        ll parentA = findUParent(a);
        ll parentB = findUParent(b);
        if(parentA != parentB){
            if(size[parentA] < size[parentB]){
                par[parentA] = parentB;
                size[parentB] += size[parentA];
            }else{
                par[parentB] = parentA;
                size[parentA] += size[parentB];
            }
        }
    }
};

ll solve(ll t, ll T)
{
    ll n;
    cin >> n;
    
    vector<vector<ll>> adj(n);
    vector<ll> degree(n, 0);
    ll a, b;
    for (int i = 0; i < (n-1); i++){
        cin >> a >> b;
        a--;
        b--;
        adj[a].push_back(b);
        adj[b].push_back(a);
        degree[a]++;
        degree[b]++;
    }

    if(n == 2){
        cout << 0 << nl;
        return 0;
    }

    
    ll maxDegree = 1;
    for(int i = 0; i < n; i++)
        maxDegree = max(maxDegree, degree[i]);
    
    vector<pair<ll,ll>> indicesWithMaxDegree;
    for(int i = 0; i < n; i++){
        if(degree[i] == maxDegree){
            ll cnt = 0;
            for(auto it:adj[i]){
                if(degree[it] == maxDegree){
                    cnt++;
                }
            }
            indicesWithMaxDegree.push_back({cnt, i});
        }
    }

    sort(indicesWithMaxDegree.begin(), indicesWithMaxDegree.end());

    ll index = indicesWithMaxDegree[0].second;
    set<ll> rem;
    rem.insert(index);

    for(auto it:adj[index])
        degree[it]--;

    maxDegree = -1;
    ll index2 = -1;
    for(int i = 0; i < n; i++){
        if(i == index)
            continue;
        if(degree[i] > maxDegree){
            maxDegree = degree[i];
            index2 = i;
        }
    }
    
    rem.insert(index2);

    DSU dsu(n);
    for(int i = 0; i < n; i++){
        for(auto it:adj[i]){
            if(i < it && rem.find(i) == rem.end() && rem.find(it) == rem.end()){
                dsu.makeUnion(i, it);
            }
        }
    }

    set<ll> parents;
    for(int i = 0; i < n; i++){
        if(rem.find(i) == rem.end()){
            parents.insert(dsu.findUParent(i));
        }
    }

    cout << parents.size() << nl;

    return 0;
}

int main()
{
    std::ios_base::sync_with_stdio(false);
    std::cin.tie(nullptr);
    ll T = 1;
    cin >> T;
    for(int t = 1; t <= T; t++)
        solve(t, T);
    return 0;
}

Solution 2 (runtime: 499 ms):
#include <bits/stdc++.h>
using namespace std;

/* clang-format off */
 
/* TYPES  */
#define ll long long
#define pii pair<int, int>
#define pll pair<long long, long long>
#define vi vector<int>
#define vll vector<long long>
#define mll map<long long, long long>
#define sll set<long long>
#define sc set<char>
#define ff first
#define ss second
 
/* FUNCTIONS */
#define f(i,s,e) for(long long int i=s;i<e;i++)
#define cf(i,s,e) for(long long int i=s;i<=e;i++)
#define rf(i,e,s) for(long long int i=e-1;i>=s;i--)
#define pb push_back
#define eb emplace_back
 
/* PRINTS */
template <class T>
void print_v(vector<T> &v) { cout<<  "{"; for (auto x : v) cout<<  x << ","; cout<<  "\b}"; }
 
/* UTILS */
#define MOD 1000000007
#define PI 3.1415926535897932384626433832795
#define read(type) readInt<type>()
ll min(ll a,int b) { if (a<b) return a; return b; }
ll min(int a,ll b) { if (a<b) return a; return b; }
ll max(ll a,int b) { if (a>b) return a; return b; }
ll max(int a,ll b) { if (a>b) return a; return b; }
ll gcd(ll a,ll b) { if (b==0) return a; return gcd(b, a%b); }
ll lcm(ll a,ll b) { return a/gcd(a,b)*b; }
string to_upper(string a) { for (int i=0;i<(int)a.size();++i) if (a[i]>='a' && a[i]<='z') a[i]-='a'-'A'; return a; }
string to_lower(string a) { for (int i=0;i<(int)a.size();++i) if (a[i]>='A' && a[i]<='Z') a[i]+='a'-'A'; return a; }
bool prime(ll a) { if (a==1) return 0; for (int i=2;i<=round(sqrt(a));++i) if (a%i==0) return 0; return 1; }
void yes() { cout<<"YES\n"; }
void no() { cout<<"NO\n"; }
 
/*  All Required define Pre-Processors and typedef Constants */
typedef long int int32;
typedef unsigned long int uint32;
typedef long long int int64;
typedef unsigned long long int  uint64;

/* clang-format on */
void solve()
{
    ll n;
    cin >> n;
    vector<set<ll>> graph(n);
    map<ll, ll> mp;
    f(i, 0, n-1)
    {
        ll x, y;
        cin >> x >> y;
        graph[x - 1].insert(y - 1);
        graph[y - 1].insert(x - 1);
    }
    f(i, 0, n)
    {
        if(mp.find(graph[i].size())==mp.end()){
            mp[graph[i].size()]=0;
        }
        mp[graph[i].size()]++;
    }
    
    ll ans = 0;
    f(i, 0, n)
    {
        ll temp = graph[i].size();
        mp[graph[i].size()]--;
        if (mp[graph[i].size()] == 0)
        {
            mp.erase(graph[i].size());
        }
        for (auto &val : graph[i])
        {
            //cout<<graph[val].size()<<" ";
            mp[graph[val].size()]--;
            if (mp[graph[val].size()] == 0)
            {
                mp.erase(graph[val].size());
            }
            if (mp.find(graph[val].size() - 1) == mp.end())
            {
                mp[graph[val].size() - 1] = 0;
            }
            mp[graph[val].size() - 1]++;
        }
        if((*mp.rbegin()).first==0){
            ans=max(temp-1,ans);
        }else{
            ans = max(ans, temp + (*mp.rbegin()).first - 1);
        }
        //cout<<ans<<endl;
        //cout<<temp<<" "<<(*mp.rbegin()).first<<endl;
        for (auto &val : graph[i])
        {
            mp[graph[val].size() - 1]--;
            if (mp[graph[val].size()-1] == 0)
            {
                mp.erase(graph[val].size()-1);
            }
            if (mp.find(graph[val].size()) == mp.end())
            {
                mp[graph[val].size()] = 0;
            }
            mp[graph[val].size()]++;
        }
        if (mp.find(graph[i].size()) == mp.end())
            {
                mp[graph[i].size()] = 0;
            }
        mp[graph[i].size()]++;
        // for(auto &[key,val]:mp){
        //     cout<<key<<" "<<val<<endl;
        // }
        // cout<<endl;
    }
    cout<<ans<<endl;
}

/* Main()  function */
int main()
{

#ifdef LOCAL
    freopen("input.txt", "rt", stdin);
    freopen("output.txt", "wt", stdout);
#endif
    ios::sync_with_stdio(0);
    cin.tie(0);
    ll t;
    cin >> t;
    while (t--)
    {
        solve();
    }
    return 0;
}