Problem: 2063C - Remove Exactly Two
Language: python

Solution 1 (runtime: 1186 ms):
import os,sys,random,threading
#sys.exit() 退出程序
#sys.setrecursionlimit(10**6) #调整栈空间
from random import randint,choice,shuffle
#randint(a,b)从[a,b]范围随机选择一个数
#choice(seq)seq可以是一个列表,元组或字符串,从seq中随机选取一个元素
#shuffle(x)将一个可变的序列x中的元素打乱
from copy import deepcopy
from io import BytesIO,IOBase
from types import GeneratorType
from functools import lru_cache,reduce
#reduce(op,迭代对象)
from bisect import bisect_left,bisect_right
#bisect_left(x) 大于等于x的第一个下标
#bisect_right(x) 大于x的第一个下标
from collections import Counter,defaultdict,deque
from itertools import accumulate,combinations,permutations
#accumulate(a)用a序列生成一个累积迭代器，一般list化前面放个[0]做前缀和用
#combinations(a,k)a序列选k个 组合迭代器
#permutations(a,k)a序列选k个 排列迭代器
from heapq import  heapify,heappop,heappush
#heapify将列表转为堆
from typing import Generic,Iterable,Iterator,TypeVar,Union,List
from string import ascii_lowercase,ascii_uppercase,digits
#小写字母，大写字母，十进制数字
from math import ceil,floor,sqrt,pi,factorial,gcd,log,log10,log2,inf
#ceil向上取整，floor向下取整 ，sqrt开方 ，factorial阶乘
from decimal import Decimal,getcontext
#Decimal(s) 实例化Decimal对象,一般使用字符串
#getcontext().prec=100 修改精度
from sys import stdin, stdout, setrecursionlimit
input = lambda: sys.stdin.readline().rstrip("\r\n")
MI = lambda :map(int,input().split())
li = lambda :list(MI())
ii = lambda :int(input())
mod = int(1e9 + 7) #998244353
inf = 1<<60
py = lambda :print("YES")
pn = lambda :print("NO")
DIRS = [(0, 1), (1, 0), (0, -1), (-1, 0)]  # 右下左上
DIRS8 = [(0, 1), (1, 1), (1, 0), (1, -1), (0, -1), (-1, -1), (-1, 0),(-1, 1)]  # →↘↓↙←↖↑↗




class SortedList:
    def __init__(self, iterable=[], _load=200):
        """Initialize sorted list instance."""
        values = sorted(iterable)
        self._len = _len = len(values);self._load = _load
        self._lists = _lists = [values[i:i + _load] for i in range(0, _len, _load)]
        self._list_lens = [len(_list) for _list in _lists]
        self._mins = [_list[0] for _list in _lists]
        self._fen_tree = [];self._rebuild = True
    def _fen_build(self):
        self._fen_tree[:] = self._list_lens
        _fen_tree = self._fen_tree
        for i in range(len(_fen_tree)):
            if i | i + 1 < len(_fen_tree):
                _fen_tree[i | i + 1] += _fen_tree[i]
        self._rebuild = False
    def _fen_update(self, index, value):
        if not self._rebuild:
            _fen_tree = self._fen_tree
            while index < len(_fen_tree):
                _fen_tree[index] += value
                index |= index + 1
    def _fen_query(self, end):
        if self._rebuild:self._fen_build()
        _fen_tree = self._fen_tree;x = 0
        while end:
            x += _fen_tree[end - 1]
            end &= end - 1
        return x
    def _fen_findkth(self, k):
        _list_lens = self._list_lens
        if k < _list_lens[0]:return 0, k
        if k >= self._len - _list_lens[-1]:
            return len(_list_lens) - 1, k + _list_lens[-1] - self._len
        if self._rebuild:self._fen_build()
        _fen_tree = self._fen_tree
        idx = -1
        for d in reversed(range(len(_fen_tree).bit_length())):
            right_idx = idx + (1 << d)
            if right_idx < len(_fen_tree) and k >= _fen_tree[right_idx]:
                idx = right_idx
                k -= _fen_tree[idx]
        return idx + 1, k
    def _delete(self, pos, idx):
        _lists = self._lists
        _mins = self._mins
        _list_lens = self._list_lens
        self._len -= 1
        self._fen_update(pos, -1)
        del _lists[pos][idx]
        _list_lens[pos] -= 1
        if _list_lens[pos]:_mins[pos] = _lists[pos][0]
        else:
            del _lists[pos]
            del _list_lens[pos]
            del _mins[pos]
            self._rebuild = True
    def _loc_left(self, value):
        if not self._len:return 0, 0
        _lists = self._lists
        _mins = self._mins
        lo, pos = -1, len(_lists) - 1
        while lo + 1 < pos:
            mi = (lo + pos) >> 1
            if value <= _mins[mi]:pos = mi
            else:lo = mi
        if pos and value <= _lists[pos - 1][-1]:pos -= 1
        _list = _lists[pos]
        lo, idx = -1, len(_list)
        while lo + 1 < idx:
            mi = (lo + idx) >> 1
            if value <= _list[mi]:idx = mi
            else:lo = mi
        return pos, idx
    def _loc_right(self, value):
        if not self._len:return 0, 0
        _lists = self._lists
        _mins = self._mins
        pos, hi = 0, len(_lists)
        while pos + 1 < hi:
            mi = (pos + hi) >> 1
            if value < _mins[mi]:hi = mi
            else:pos = mi
        _list = _lists[pos]
        lo, idx = -1, len(_list)
        while lo + 1 < idx:
            mi = (lo + idx) >> 1
            if value < _list[mi]:idx = mi
            else:lo = mi
        return pos, idx
    def add(self, value):
        _load = self._load
        _lists = self._lists
        _mins = self._mins
        _list_lens = self._list_lens
        self._len += 1
        if _lists:
            pos, idx = self._loc_right(value)
            self._fen_update(pos, 1)
            _list = _lists[pos]
            _list.insert(idx, value)
            _list_lens[pos] += 1
            _mins[pos] = _list[0]
            if _load + _load < len(_list):
                _lists.insert(pos + 1, _list[_load:])
                _list_lens.insert(pos + 1, len(_list) - _load)
                _mins.insert(pos + 1, _list[_load])
                _list_lens[pos] = _load
                del _list[_load:]
                self._rebuild = True
        else:
            _lists.append([value])
            _mins.append(value)
            _list_lens.append(1)
            self._rebuild = True
    def discard(self, value):
        _lists = self._lists
        if _lists:
            pos, idx = self._loc_right(value)
            if idx and _lists[pos][idx - 1] == value:
                self._delete(pos, idx - 1)
    def remove(self, value):
        _len = self._len
        self.discard(value)
        if _len == self._len:raise Valueerror('{0!r} not in list'.format(value))
    def pop(self, index=-1):
        pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
        value = self._lists[pos][idx]
        self._delete(pos, idx)
        return value
    def bisect_left(self, value):
        pos, idx = self._loc_left(value)
        return self._fen_query(pos) + idx
    def bisect_right(self, value):
        pos, idx = self._loc_right(value)
        return self._fen_query(pos) + idx
    def count(self, value):
        return self.bisect_right(value) - self.bisect_left(value)
    def __len__(self):
        return self._len
    def __getitem__(self, index):
        pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
        return self._lists[pos][idx]
    def __delitem__(self, index):
        pos, idx = self._fen_findkth(self._len + index if index < 0 else index)
        self._delete(pos, idx)
    def __contains__(self, value):
        _lists = self._lists
        if _lists:
            pos, idx = self._loc_left(value)
            return idx < len(_lists[pos]) and _lists[pos][idx] == value
        return False
    def __iter__(self):
        return (value for _list in self._lists for value in _list)
    def __reversed__(self):
        return (value for _list in reversed(self._lists) for value in reversed(_list))
    def __repr__(self):
        return 'SortedList({0})'.format(list(self))


t=ii()

for _ in range(t):
    n=ii()
    g=[[] for _ in range(n)]
    deg=[0]*(n)
    for _ in range(n-1):
        u,v=li()
        u-=1
        v-=1
        g[u].append(v)
        g[v].append(u)
        deg[u]+=1
        deg[v]+=1
    sl=SortedList(deg+[0])
    res=0
    for i in range(n):
        sl.remove(deg[i])
        for v in g[i]:
            sl.remove(deg[v])
            deg[v]-=1
            sl.add(deg[v])
        res=max(res,deg[i]+sl[-1]-1)
        sl.add(deg[i])
        for v in g[i]:
            sl.remove(deg[v])
            deg[v]+=1
            sl.add(deg[v])
    print(res)






Solution 2 (runtime: 1374 ms):
from collections import defaultdict

t = int(input())

def find_max_edges(graph):
    max_us = []
    max_n = -1
    for u in graph:
        if len(graph[u]) > max_n:
            max_us, max_n = [u], len(graph[u])
        elif len(graph[u]) == max_n:
            max_us.append(u)
    return max_us, max_n

for _ in range(t):
    g = defaultdict(list)
    g_lengths = defaultdict(int)
    n = int(input())
    for i in range(n - 1):
        u, v = input().split(' ')
        g[u].append(v)
        g[v].append(u)
        g_lengths[u] += 1
        g_lengths[v] += 1

    us, n = find_max_edges(g)
    max_2nd_edges = 0
    for u in us:
        u_adj = set(g[u])
        for node in g_lengths:
            if node == u:
                continue
            max_2nd_edges = max(g_lengths[node] - (1 if node in u_adj else 0), max_2nd_edges)
        if max_2nd_edges == n:
            break

    components = 1
    components += n - 1
    components += max_2nd_edges - 1
    print(components)