/** * author: nok0 * created: 2020.12.30 21:50:20 **/ #ifdef LOCAL #define _GLIBCXX_DEBUG #endif #include using namespace std; #pragma region Macros #define FOR(i, l, r) for(int i = (l); i < (r); ++i) #define REP(i, n) FOR(i, 0, n) #define REPS(i, n) FOR(i, 1, n + 1) #define RFOR(i, l, r) for(int i = (l); i >= (r); --i) #define RREP(i, n) RFOR(i, n - 1, 0) #define RREPS(i, n) RFOR(i, n, 1) #define pb push_back #define eb emplace_back #define SZ(x) ((int)(x).size()) #define all(x) (x).begin(), (x).end() #define rall(x) (x).rbegin(), (x).rend() template using V = vector; template using VV = V>; using ll = long long; using ld = long double; using pii = pair; using pll = pair; #define VEC(type, name, size) \ V name(size); \ INPUT(name) #define VVEC(type, name, h, w) \ VV name(h, V(w)); \ INPUT(name) #define INT(...) \ int __VA_ARGS__; \ INPUT(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ INPUT(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ INPUT(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ INPUT(__VA_ARGS__) #define DOUBLE(...) \ DOUBLE __VA_ARGS__; \ INPUT(__VA_ARGS__) #define LD(...) \ LD __VA_ARGS__; \ INPUT(__VA_ARGS__) void scan(int &a) { cin >> a; } void scan(long long &a) { cin >> a; } void scan(char &a) { cin >> a; } void scan(double &a) { cin >> a; } void scan(long double &a) { cin >> a; } void scan(char a[]) { scanf("%s", a); } void scan(string &a) { cin >> a; } template void scan(V &); template void scan(pair &); template void scan(V &a) { for(auto &i : a) scan(i); } template void scan(pair &p) { scan(p.first); scan(p.second); } template void scan(T &a) { cin >> a; } void INPUT() {} template void INPUT(Head &head, Tail &... tail) { scan(head); INPUT(tail...); } template inline void print(T x) { cout << x << '\n'; } template istream &operator>>(istream &is, pair &p) { is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const pair &p) { os << p.first << " " << p.second; return os; } template istream &operator>>(istream &is, vector &v) { for(T &in : v) is >> in; return is; } template ostream &operator<<(ostream &os, const V &v) { REP(i, SZ(v)) { if(i) os << " "; os << v[i]; } return os; } //debug template void view(const V &v) { cerr << "{ "; for(const auto &e : v) { cerr << e << ", "; } cerr << "\b\b }"; } template void view(const VV &vv) { cerr << "{\n"; for(const auto &v : vv) { cerr << "\t"; view(v); cerr << "\n"; } cerr << "}"; } template void view(const V> &v) { cerr << "{\n"; for(const auto &c : v) cerr << "\t(" << c.first << ", " << c.second << ")\n"; cerr << "}"; } template void view(const map &m) { cerr << "{\n"; for(auto &t : m) cerr << "\t[" << t.first << "] : " << t.second << "\n"; cerr << "}"; } template void view(const pair &p) { cerr << "(" << p.first << ", " << p.second << ")"; } template void view(const set &s) { cerr << "{ "; for(auto &t : s) { view(t); cerr << ", "; } cerr << "\b\b }"; } template void view(T e) { cerr << e; } #ifdef LOCAL void debug_out() {} template void debug_out(Head H, Tail... T) { view(H); cerr << ", "; debug_out(T...); } #define debug(...) \ do { \ cerr << __LINE__ << " [" << #__VA_ARGS__ << "] : ["; \ debug_out(__VA_ARGS__); \ cerr << "\b\b]\n"; \ } while(0) #else #define debug(...) (void(0)) #endif template V press(V &x) { V res = x; sort(all(res)); res.erase(unique(all(res)), res.end()); REP(i, SZ(x)) { x[i] = lower_bound(all(res), x[i]) - res.begin(); } return res; } template inline bool chmin(T &a, T b) { if(a > b) { a = b; return true; } return false; } template inline bool chmax(T &a, T b) { if(a < b) { a = b; return true; } return false; } inline void Yes(bool b = true) { cout << (b ? "Yes" : "No") << '\n'; } inline void YES(bool b = true) { cout << (b ? "YES" : "NO") << '\n'; } inline void err(bool b = true) { if(b) { cout << -1 << '\n'; exit(0); } } template inline void fin(bool b = true, T e = 0) { if(b) { cout << e << '\n'; exit(0); } } template T divup(T x, T y) { return (x + (y - 1)) / y; } template T pow(T a, long long n, T e = 1) { T ret = e; while(n) { if(n & 1) ret *= a; a *= a; n >>= 1; } return ret; } struct iofast { iofast() { ios::sync_with_stdio(false); cin.tie(nullptr); cout << fixed << setprecision(15); } } iofast_; const int inf = 1e9; const ll INF = 1e18; #pragma endregion struct Edge { int to; long long cost; Edge() = default; Edge(int to_, long long cost_) : to(to_), cost(cost_) {} bool operator<(const Edge &a) const { return cost < a.cost; } bool operator>(const Edge &a) const { return cost > a.cost; } friend std::ostream &operator<<(std::ostream &s, Edge &a) { s << "to:" << a.to << " cost:" << a.cost; return s; } }; struct Graph : private std::vector> { using std::vector>::vector; using std::vector>::operator[]; using std::vector>::operator=; using std::vector>::size; using std::vector>::resize; using std::vector>::clear; using std::vector>::push_back; using std::vector>::emplace_back; using std::vector>::begin; using std::vector>::end; private: const long long INF = 1e18; std::vector> edges; std::vector> cedges; bool isweight = true, lcainit_done = false; std::vector> parent; std::vector lcadist; template inline bool chmax(T &a, T b) { if(a < b) { a = b; return true; } return false; } template inline bool chmin(T &a, T b) { if(a > b) { a = b; return true; } return false; } public: void read(int e = -1, bool weight = 0, bool directed = false, int idx = 1) { if(e == -1) e = size() - 1; if(!weight) isweight = false; while(e--) { int u, v; long long cost = 1; std::cin >> u >> v; if(weight) std::cin >> cost; u -= idx, v -= idx; (*this)[u].emplace_back(v, cost); edges.emplace_back(u, v); cedges.emplace_back(u, v, cost); if(!directed) (*this)[v].emplace_back(u, cost); } } void add(int u, int v, int cost = -1, bool directed = false, int idx = 0) { if(cost != -1) isweight = true; else cost = 1; u -= idx, v -= idx; (*this)[u].emplace_back(v, cost); edges.emplace_back(u, v); cedges.emplace_back(u, v, cost); if(!directed) (*this)[v].emplace_back(u, cost); } std::vector Dijkstra(int s) { std::vector dist(size(), INF); if(isweight) { using pll = std::pair; std::priority_queue, std::greater> que; dist[s] = 0; que.push(pll(0, s)); while(!que.empty()) { pll p = que.top(); que.pop(); int v = p.second; if(dist[v] < p.first) continue; for(int i = 0; i < int((*this)[v].size()); i++) { Edge e = (*this)[v][i]; if(dist[e.to] > dist[v] + e.cost) { dist[e.to] = dist[v] + e.cost; que.push(pll(dist[e.to], e.to)); } } } return dist; } else { std::vector dist(size(), INF); std::queue que; dist[s] = 0; que.push(s); while(!que.empty()) { int v = que.front(); que.pop(); for(auto nv : (*this)[v]) { if(dist[nv.to] != INF) continue; dist[nv.to] = dist[v] + 1; que.push(nv.to); } } return dist; } } std::vector BellmanFord(int s) { int n = size(); std::vector res(n, INF); res[s] = 0; for(int loop = 0; loop < n - 1; loop++) { for(int v = 0; v < n; v++) { if(res[v] == INF) continue; for(auto &e : (*this)[v]) { res[e.to] = std::min(res[e.to], res[v] + e.cost); } } } std::queue que; std::vector chk(n, 0); for(int v = 0; v < n; v++) { if(res[v] == INF) continue; for(auto &e : (*this)[v]) { if(res[e.to] > res[v] + e.cost) { res[e.to] = res[v] + e.cost; if(!chk[e.to]) { que.push(e.to); chk[e.to] = 1; } } } } while(!que.empty()) { int now = que.front(); que.pop(); for(auto &e : (*this)[now]) { if(!chk[e.to]) { chk[e.to] = 1; que.push(e.to); } } } for(int i = 0; i < n; i++) if(chk[i]) res[i] = -INF; return res; } std::vector> WarshallFloyd() { int n = size(); std::vector dist(n, std::vector(n, INF)); for(int i = 0; i < n; i++) { dist[i][i] = 0; } for(int i = 0; i < n; i++) { for(auto e : (*this)[i]) { chmin(dist[i][e.to], e.cost); } } for(int k = 0; k < n; k++) for(int i = 0; i < n; i++) for(int j = 0; j < n; j++) chmin(dist[i][j], dist[i][k] + dist[k][j]); return dist; } int lca(int u, int v) { std::function dfs = [&](int now, int per, int d) { parent[0][now] = per; lcadist[now] = d; for(auto &e : (*this)[now]) if(e.to != per) dfs(e.to, now, d + 1); }; if(!lcainit_done) { lcainit_done = true; int n = size(); int k = 1; while((1 << k) < n) k++; parent.assign(k, std::vector(n, -1)); lcadist.assign(n, -1); dfs(0, -1, 0); for(int i = 0; i < k - 1; i++) for(int v = 0; v < n; v++) { if(parent[i][v] < 0) parent[i + 1][v] = -1; else parent[i + 1][v] = parent[i][parent[i][v]]; } } if(lcadist[u] < lcadist[v]) std::swap(u, v); int k = parent.size(); for(int i = 0; i < k; i++) if((lcadist[u] - lcadist[v]) >> i & 1) u = parent[i][u]; if(u == v) return u; for(int i = k - 1; i >= 0; i--) if(parent[i][u] != parent[i][v]) u = parent[i][u], v = parent[i][v]; return parent[0][u]; } int dist(int u, int v) { return lcadist[u] + lcadist[v] - 2 * lcadist[lca(u, v)]; } std::vector tsort() { std::vector res; int n = size(); std::vector used(n, 0); bool notDAG = false; std::function dfs = [&](int k) { if(notDAG) return; if(used[k]) { if(used[k] == 1) notDAG = true; return; } used[k] = 1; for(auto &e : (*this)[k]) dfs(e.to); used[k] = 2; res.push_back(k); }; for(int i = 0; i < n; i++) dfs(i); if(notDAG) return std::vector{}; std::reverse(res.begin(), res.end()); return res; } bool isDAG() { return tsort().size(); } std::vector height() { std::vector fir(size(), -1), sec(size(), -1); std::function dfs = [&](int now, int par) { long long f = 0, s = 0; for(auto &e : (*this)[now]) { if(e.to == par) continue; s = std::max(s, dfs(e.to, now) + e.cost); if(f < s) std::swap(f, s); } sec[now] = s; return fir[now] = f; }; std::function solve = [&](int now, int par, long long parf, long long pars, long long parcost) { if(fir[now] + parcost == parf) parf = pars; sec[now] = std::max(sec[now], parf + parcost); if(fir[now] < sec[now]) std::swap(fir[now], sec[now]); for(auto &e : (*this)[now]) { if(e.to == par) continue; solve(e.to, now, fir[now], sec[now], e.cost); } return; }; dfs(0, -1); solve(0, -1, -1, -1, -1); return fir; } //{二部グラフか、二部グラフのどちらに属するかを0か1で表した配列} std::pair> isBiparate() { std::vector colors(size()), ret(size()); std::function dfs = [&](int now, int col) { colors[now] = col; if(col == 1) ret[now] = 1; for(auto &e : (*this)[now]) { if(col == colors[e.to]) return false; if(colors[e.to] == 0 and !dfs(e.to, -col)) return false; } return true; }; return std::pair(dfs(0, 1), ret); } //{{端点、端点},直径の大きさ} std::pair, long long> Diameter() { auto vec = Dijkstra(0); int v1 = -1, v2 = -1; long long dia = -1; for(int i = 0; i < int(size()); i++) if(chmax(dia, vec[i])) v1 = i; vec = Dijkstra(v1); dia = -1; for(int i = 0; i < int(size()); i++) if(chmax(dia, vec[i])) v2 = i; std::pair, long long> res = {{v1, v2}, dia}; return res; } //最小全域木 long long Prim() { long long res = 0; std::priority_queue, std::greater> que; for(auto &e : (*this)[0]) que.push(e); std::vector chk(size(), 0); chk[0] = 1; while(!que.empty()) { auto e = que.top(); que.pop(); if(chk[e.to]) continue; res += e.cost; chk[e.to] = 1; for(auto &e : (*this)[e.to]) { que.push(e); } } return res; } //重心 std::vector Centroid() { int n = size(); std::vector centroid, sz(n); std::function dfs = [&](int now, int per) { sz[now] = 1; bool is_centroid = true; for(auto &e : (*this)[now]) if(e.to != per) { dfs(e.to, now); sz[now] += sz[e.to]; if(sz[e.to] > n / 2) is_centroid = false; } if(n - sz[now] > n / 2) is_centroid = false; if(is_centroid) centroid.push_back(now); }; return centroid; } //pair{橋(pairの配列),関節点(intの配列)} std::pair>, std::vector> lowlink() { int n = size(), k = 0; std::vector used(n, 0), order(n, 0), low(n, 0), articulation; std::vector> bridge; std::function dfs = [&](int idx, int k, int par) { used[idx] = 1; order[idx] = k++; low[idx] = order[idx]; bool is_articulation = false, neko = false; int cnt = 0; for(auto &e : (*this)[idx]) { if(e.to == par and !std::exchange(neko, true)) { continue; } if(!used[e.to]) { ++cnt; k = dfs(e.to, k, idx); low[idx] = std::min(low[idx], low[e.to]); is_articulation |= par >= 0 and low[e.to] >= order[idx]; if(order[idx] < low[e.to]) { auto p = std::minmax(idx, e.to); bridge.emplace_back(p.first, p.second); } } else { low[idx] = std::min(low[idx], order[e.to]); } } is_articulation |= par == -1 and cnt > 1; if(is_articulation) articulation.push_back(idx); return k; }; for(int i = 0; i < n; i++) if(!used[i]) k = dfs(i, k, -1); return std::pair(bridge, articulation); } //根から葉への有向グラフ Graph RootToLeaf(int root = 0) { Graph res(size()); std::vector chk(size(), 0); chk[root] = 1; std::function dfs = [&](int now) { for(auto &e : (*this)[now]) { if(chk[e.to] == 1) continue; chk[e.to] = 1; res.add(now, e.to, e.cost, 1, 0); dfs(e.to); } }; dfs(root); return res; } //葉から根への有向グラフ Graph LeafToRoot(int root = 0) { Graph res(size()); std::vector chk(size(), 0); chk[root] = 1; std::function dfs = [&](int now) { for(auto &e : (*this)[now]) { if(chk[e.to] == 1) continue; chk[e.to] = 1; res.add(e.to, now, e.cost, 1, 0); dfs(e.to); } }; dfs(root); return res; } //辺(from,to)を返す std::vector> getedges() { return edges; } //辺(from,to,cost)を返す std::vector> getwedges() { return cedges; } }; struct SCC { private: const Graph &g; Graph ng, rg; std::vector compo, order, used; void dfs(int now) { if(used[now]) return; used[now] = true; for(auto &e : ng[now]) dfs(e.to); order.emplace_back(now); } void rdfs(int now, int count) { if(compo[now] != -1) return; compo[now] = count; for(auto &e : rg[now]) rdfs(e.to, count); } public: SCC(Graph &g_) : g(g_) { int n = g.size(); ng.resize(n); rg.resize(n); compo.assign(n, -1); used.resize(n); for(int i = 0; i < n; i++) { for(auto &e : g[i]) { ng[i].emplace_back(e.to, 1); rg[e.to].emplace_back(i, 1); } } } int operator[](int k) { return compo[k]; } Graph build() { Graph ret; for(int i = 0; i < (int)ng.size(); i++) dfs(i); reverse(order.begin(), order.end()); int group = 0; for(auto i : order) if(compo[i] == -1) rdfs(i, group), group++; ret.resize(group); for(int i = 0; i < (int)g.size(); i++) { for(auto &e : g[i]) { int s = compo[i], t = compo[e.to]; if(s != t) ret[s].emplace_back(t, 1); } } return std::move(ret); } }; struct HeavyLightDecomposition { public: std::vector sz, in, out, head, rev, par; private: Graph &g; void dfs_sz(int v, int p = -1) { par[v] = p; if(!g[v].empty() and g[v].front().to == p) std::swap(g[v].front(), g[v].back()); for(auto &e : g[v]) { if(e.to == p) continue; dfs_sz(e.to, v); sz[v] += sz[e.to]; if(sz[g[v].front().to] < sz[e.to]) std::swap(g[v].front(), e); } } void dfs_hld(int v, int &t, int p = -1) { in[v] = t++; rev[in[v]] = v; for(auto &e : g[v]) { if(e.to == p) continue; head[e.to] = (g[v].front().to == e.to ? head[v] : e.to); dfs_hld(e.to, t, v); } out[v] = t; } void build(int root = 0) { dfs_sz(root); int t = 0; head[root] = root; dfs_hld(root, t); } public: HeavyLightDecomposition(Graph &g_, int root = 0) : g(g_) { int n = g.size(); sz.assign(n, 1); in.resize(n); out.resize(n); head.resize(n); rev.resize(n); par.resize(n); build(root); } //頂点vから0方向にk遡った頂点を返す int la(int v, int k) { while(true) { int u = head[v]; if(in[v] - k >= in[u]) return rev[in[v] - k]; k -= in[v] - in[u] + 1; v = par[u]; } } int lca(int u, int v) { for(;; v = par[head[v]]) { if(in[u] > in[v]) std::swap(u, v); if(head[u] == head[v]) return u; } } //u,v:頂点 ti:単位元 q:列にたいするクエリ f:列のmerge template T query(int u, int v, const T &ti, const Q &q, const F &f, bool edge = false) { T l = ti, r = ti; for(;; v = par[head[v]]) { if(in[u] > in[v]) std::swap(u, v), std::swap(l, r); if(head[u] == head[v]) break; l = f(q(in[head[v]], in[v] + 1), l); } return f(f(q(in[u] + edge, in[v] + 1), l), r); } //u,v:頂点 q:更新クエリ template void add(int u, int v, const Q &q, bool edge = false) { for(;; v = par[head[v]]) { if(in[u] > in[v]) std::swap(u, v); if(head[u] == head[v]) break; q(in[head[v]], in[v] + 1); } q(in[u] + edge, in[v] + 1); } std::pair subtree(int v, bool edge = false) { return std::pair(in[v] + edge, out[v]); } }; int main() { INT(n); Graph G(n); G.read(); V<> deg; REP(i, n) { deg.pb(SZ(G[i])); } if(*max_element(all(deg)) <= 2) { Yes(1); return 0; } int center = -1; REP(i, n) { if(deg[i] > 2) { if(center == -1) center = i; else { Yes(0); return 0; } } } Graph H = G.RootToLeaf(center); auto dfs = [&H](auto self, int now) -> int { int ret = 0; if(SZ(H[now]) > 1) return -inf; for(auto &[to, ig] : H[now]) ret += self(self, to) + 1; return ret; }; int tmp = dfs(dfs, H[center][0].to); bool ok = true; for(auto &[to, ig] : H[center]) if(tmp != dfs(dfs, to)) ok = false; Yes(ok); return 0; }