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:heavy_check_mark: graph/ds/tree_monoid.hpp

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#pragma once

#include "ds/segtree/segtree.hpp"

#include "graph/tree.hpp"

#include "alg/monoid/monoid_reverse.hpp"


template <typename TREE, typename Monoid, bool edge>
struct Tree_Monoid {
  using MX = Monoid;
  using X = typename MX::value_type;
  TREE &tree;
  int N;
  SegTree<MX> seg;
  SegTree<Monoid_Reverse<MX>> seg_r;

  Tree_Monoid(TREE &tree) : tree(tree), N(tree.N) {
    build([](int i) -> X { return MX::unit(); });
  }

  Tree_Monoid(TREE &tree, vc<X> &dat) : tree(tree), N(tree.N) {
    build([&](int i) -> X { return dat[i]; });
  }

  template <typename F>
  Tree_Monoid(TREE &tree, F f) : tree(tree), N(tree.N) {
    build(f);
  }

  template <typename F>
  void build(F f) {
    if (!edge) {
      auto f_v = [&](int i) -> X { return f(tree.V[i]); };
      seg.build(N, f_v);
      if constexpr (!MX::commute) { seg_r.build(N, f_v); }
    } else {
      auto f_e = [&](int i) -> X {
        return (i == 0 ? MX::unit() : f(tree.v_to_e(tree.V[i])));
      };
      seg.build(N, f_e);
      if constexpr (!MX::commute) { seg_r.build(N, f_e); }
    }
  }

  void set(int i, X x) {
    if constexpr (edge) i = tree.e_to_v(i);
    i = tree.LID[i];
    seg.set(i, x);
    if constexpr (!MX::commute) seg_r.set(i, x);
  }

  void multiply(int i, X x) {
    if constexpr (edge) i = tree.e_to_v(i);
    i = tree.LID[i];
    seg.multiply(i, x);
    if constexpr (!MX::commute) seg_r.multiply(i, x);
  }

  X prod_path(int u, int v) {
    auto pd = tree.get_path_decomposition(u, v, edge);
    X val = MX::unit();
    for (auto &&[a, b]: pd) { val = MX::op(val, get_prod(a, b)); }
    return val;
  }

  // uv path 上で prod_path(u, x) が check を満たす最後の x

  // なければ （つまり path(u,u) が ng ）-1

  template <class F>
  int max_path(F check, int u, int v) {
    if constexpr (edge) return max_path_edge(check, u, v);
    if (!check(prod_path(u, u))) return -1;
    auto pd = tree.get_path_decomposition(u, v, edge);
    X val = MX::unit();
    for (auto &&[a, b]: pd) {
      X x = get_prod(a, b);
      if (check(MX::op(val, x))) {
        val = MX::op(val, x);
        u = (tree.V[b]);
        continue;
      }
      auto check_tmp = [&](X x) -> bool { return check(MX::op(val, x)); };
      if (a <= b) {
        // 下り

        auto i = seg.max_right(check_tmp, a);
        return (i == a ? u : tree.V[i - 1]);
      } else {
        // 上り

        int i = 0;
        if constexpr (MX::commute) i = seg.min_left(check_tmp, a + 1);
        if constexpr (!MX::commute) i = seg_r.min_left(check_tmp, a + 1);
        if (i == a + 1) return u;
        return tree.V[i];
      }
    }
    return v;
  }

  X prod_subtree(int u) {
    int l = tree.LID[u], r = tree.RID[u];
    return seg.prod(l + edge, r);
  }

  X prod_all() { return prod_subtree(tree.V[0]); }

  inline X get_prod(int a, int b) {
    if constexpr (MX::commute) {
      return (a <= b) ? seg.prod(a, b + 1) : seg.prod(b, a + 1);
    }
    return (a <= b) ? seg.prod(a, b + 1) : seg_r.prod(b, a + 1);
  }

private:
  template <class F>
  int max_path_edge(F check, int u, int v) {
    static_assert(edge);
    if (!check(MX::unit())) return -1;
    int lca = tree.lca(u, v);
    auto pd = tree.get_path_decomposition(u, lca, edge);
    X val = MX::unit();

    // climb

    for (auto &&[a, b]: pd) {
      assert(a >= b);
      X x = get_prod(a, b);
      if (check(MX::op(val, x))) {
        val = MX::op(val, x);
        u = (tree.parent[tree.V[b]]);
        continue;
      }
      auto check_tmp = [&](X x) -> bool { return check(MX::op(val, x)); };
      int i = 0;
      if constexpr (MX::commute) i = seg.min_left(check_tmp, a + 1);
      if constexpr (!MX::commute) i = seg_r.min_left(check_tmp, a + 1);
      if (i == a + 1) return u;
      return tree.parent[tree.V[i]];
    }
    // down

    pd = tree.get_path_decomposition(lca, v, edge);
    for (auto &&[a, b]: pd) {
      assert(a <= b);
      X x = get_prod(a, b);
      if (check(MX::op(val, x))) {
        val = MX::op(val, x);
        u = (tree.V[b]);
        continue;
      }
      auto check_tmp = [&](X x) -> bool { return check(MX::op(val, x)); };
      auto i = seg.max_right(check_tmp, a);
      return (i == a ? u : tree.V[i - 1]);
    }
    return v;
  }
};
#line 2 "graph/ds/tree_monoid.hpp"

#line 2 "ds/segtree/segtree.hpp"

template <class Monoid>
struct SegTree {
  using MX = Monoid;
  using X = typename MX::value_type;
  using value_type = X;
  vc<X> dat;
  int n, log, size;

  SegTree() {}
  SegTree(int n) { build(n); }
  template <typename F>
  SegTree(int n, F f) {
    build(n, f);
  }
  SegTree(const vc<X>& v) { build(v); }

  void build(int m) {
    build(m, [](int i) -> X { return MX::unit(); });
  }
  void build(const vc<X>& v) {
    build(len(v), [&](int i) -> X { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m, log = 1;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    dat.assign(size << 1, MX::unit());
    FOR(i, n) dat[size + i] = f(i);
    FOR_R(i, 1, size) update(i);
  }

  X get(int i) { return dat[size + i]; }
  vc<X> get_all() { return {dat.begin() + size, dat.begin() + size + n}; }

  void update(int i) { dat[i] = Monoid::op(dat[2 * i], dat[2 * i + 1]); }
  void set(int i, const X& x) {
    assert(i < n);
    dat[i += size] = x;
    while (i >>= 1) update(i);
  }

  void multiply(int i, const X& x) {
    assert(i < n);
    i += size;
    dat[i] = Monoid::op(dat[i], x);
    while (i >>= 1) update(i);
  }

  X prod(int L, int R) {
    assert(0 <= L && L <= R && R <= n);
    X vl = Monoid::unit(), vr = Monoid::unit();
    L += size, R += size;
    while (L < R) {
      if (L & 1) vl = Monoid::op(vl, dat[L++]);
      if (R & 1) vr = Monoid::op(dat[--R], vr);
      L >>= 1, R >>= 1;
    }
    return Monoid::op(vl, vr);
  }

  X prod_all() { return dat[1]; }

  template <class F>
  int max_right(F check, int L) {
    assert(0 <= L && L <= n && check(Monoid::unit()));
    if (L == n) return n;
    L += size;
    X sm = Monoid::unit();
    do {
      while (L % 2 == 0) L >>= 1;
      if (!check(Monoid::op(sm, dat[L]))) {
        while (L < size) {
          L = 2 * L;
          if (check(Monoid::op(sm, dat[L]))) { sm = Monoid::op(sm, dat[L++]); }
        }
        return L - size;
      }
      sm = Monoid::op(sm, dat[L++]);
    } while ((L & -L) != L);
    return n;
  }

  template <class F>
  int min_left(F check, int R) {
    assert(0 <= R && R <= n && check(Monoid::unit()));
    if (R == 0) return 0;
    R += size;
    X sm = Monoid::unit();
    do {
      --R;
      while (R > 1 && (R % 2)) R >>= 1;
      if (!check(Monoid::op(dat[R], sm))) {
        while (R < size) {
          R = 2 * R + 1;
          if (check(Monoid::op(dat[R], sm))) { sm = Monoid::op(dat[R--], sm); }
        }
        return R + 1 - size;
      }
      sm = Monoid::op(dat[R], sm);
    } while ((R & -R) != R);
    return 0;
  }

  // prod_{l<=i<r} A[i xor x]
  X xor_prod(int l, int r, int xor_val) {
    static_assert(Monoid::commute);
    X x = Monoid::unit();
    for (int k = 0; k < log + 1; ++k) {
      if (l >= r) break;
      if (l & 1) { x = Monoid::op(x, dat[(size >> k) + ((l++) ^ xor_val)]); }
      if (r & 1) { x = Monoid::op(x, dat[(size >> k) + ((--r) ^ xor_val)]); }
      l /= 2, r /= 2, xor_val /= 2;
    }
    return x;
  }
};
#line 2 "graph/tree.hpp"

#line 2 "graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (len(used_e) <= e.id) used_e.resize(e.id + 1);
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 4 "graph/tree.hpp"

// HLD euler tour をとっていろいろ。

template <typename GT>
struct Tree {
  using Graph_type = GT;
  GT &G;
  using WT = typename GT::cost_type;
  int N;
  vector<int> LID, RID, head, V, parent, VtoE;
  vc<int> depth;
  vc<WT> depth_weighted;

  Tree(GT &G, int r = 0, bool hld = 1) : G(G) { build(r, hld); }

  void build(int r = 0, bool hld = 1) {
    if (r == -1) return; // build を遅延したいとき

    N = G.N;
    LID.assign(N, -1), RID.assign(N, -1), head.assign(N, r);
    V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1);
    depth.assign(N, -1), depth_weighted.assign(N, 0);
    assert(G.is_prepared());
    int t1 = 0;
    dfs_sz(r, -1, hld);
    dfs_hld(r, t1);
  }

  void dfs_sz(int v, int p, bool hld) {
    auto &sz = RID;
    parent[v] = p;
    depth[v] = (p == -1 ? 0 : depth[p] + 1);
    sz[v] = 1;
    int l = G.indptr[v], r = G.indptr[v + 1];
    auto &csr = G.csr_edges;
    // 使う辺があれば先頭にする

    for (int i = r - 2; i >= l; --i) {
      if (hld && depth[csr[i + 1].to] == -1) swap(csr[i], csr[i + 1]);
    }
    int hld_sz = 0;
    for (int i = l; i < r; ++i) {
      auto e = csr[i];
      if (depth[e.to] != -1) continue;
      depth_weighted[e.to] = depth_weighted[v] + e.cost;
      VtoE[e.to] = e.id;
      dfs_sz(e.to, v, hld);
      sz[v] += sz[e.to];
      if (hld && chmax(hld_sz, sz[e.to]) && l < i) { swap(csr[l], csr[i]); }
    }
  }

  void dfs_hld(int v, int &times) {
    LID[v] = times++;
    RID[v] += LID[v];
    V[LID[v]] = v;
    bool heavy = true;
    for (auto &&e: G[v]) {
      if (depth[e.to] <= depth[v]) continue;
      head[e.to] = (heavy ? head[v] : e.to);
      heavy = false;
      dfs_hld(e.to, times);
    }
  }

  vc<int> heavy_path_at(int v) {
    vc<int> P = {v};
    while (1) {
      int a = P.back();
      for (auto &&e: G[a]) {
        if (e.to != parent[a] && head[e.to] == v) {
          P.eb(e.to);
          break;
        }
      }
      if (P.back() == a) break;
    }
    return P;
  }

  int heavy_child(int v) {
    int k = LID[v] + 1;
    if (k == N) return -1;
    int w = V[k];
    return (parent[w] == v ? w : -1);
  }

  int e_to_v(int eid) {
    auto e = G.edges[eid];
    return (parent[e.frm] == e.to ? e.frm : e.to);
  }
  int v_to_e(int v) { return VtoE[v]; }

  int ELID(int v) { return 2 * LID[v] - depth[v]; }
  int ERID(int v) { return 2 * RID[v] - depth[v] - 1; }

  // 目標地点へ進む個数が k

  int LA(int v, int k) {
    assert(k <= depth[v]);
    while (1) {
      int u = head[v];
      if (LID[v] - k >= LID[u]) return V[LID[v] - k];
      k -= LID[v] - LID[u] + 1;
      v = parent[u];
    }
  }
  int la(int u, int v) { return LA(u, v); }

  int LCA(int u, int v) {
    for (;; v = parent[head[v]]) {
      if (LID[u] > LID[v]) swap(u, v);
      if (head[u] == head[v]) return u;
    }
  }
  // root を根とした場合の lca

  int LCA_root(int u, int v, int root) {
    return LCA(u, v) ^ LCA(u, root) ^ LCA(v, root);
  }
  int lca(int u, int v) { return LCA(u, v); }
  int lca_root(int u, int v, int root) { return LCA_root(u, v, root); }

  int subtree_size(int v, int root = -1) {
    if (root == -1) return RID[v] - LID[v];
    if (v == root) return N;
    int x = jump(v, root, 1);
    if (in_subtree(v, x)) return RID[v] - LID[v];
    return N - RID[x] + LID[x];
  }

  int dist(int a, int b) {
    int c = LCA(a, b);
    return depth[a] + depth[b] - 2 * depth[c];
  }

  WT dist_weighted(int a, int b) {
    int c = LCA(a, b);
    return depth_weighted[a] + depth_weighted[b] - WT(2) * depth_weighted[c];
  }

  // a is in b

  bool in_subtree(int a, int b) { return LID[b] <= LID[a] && LID[a] < RID[b]; }

  int jump(int a, int b, ll k) {
    if (k == 1) {
      if (a == b) return -1;
      return (in_subtree(b, a) ? LA(b, depth[b] - depth[a] - 1) : parent[a]);
    }
    int c = LCA(a, b);
    int d_ac = depth[a] - depth[c];
    int d_bc = depth[b] - depth[c];
    if (k > d_ac + d_bc) return -1;
    if (k <= d_ac) return LA(a, k);
    return LA(b, d_ac + d_bc - k);
  }

  vc<int> collect_child(int v) {
    vc<int> res;
    for (auto &&e: G[v])
      if (e.to != parent[v]) res.eb(e.to);
    return res;
  }

  vc<int> collect_light(int v) {
    vc<int> res;
    bool skip = true;
    for (auto &&e: G[v])
      if (e.to != parent[v]) {
        if (!skip) res.eb(e.to);
        skip = false;
      }
    return res;
  }

  vc<pair<int, int>> get_path_decomposition(int u, int v, bool edge) {
    // [始点, 終点] の"閉"区間列。

    vc<pair<int, int>> up, down;
    while (1) {
      if (head[u] == head[v]) break;
      if (LID[u] < LID[v]) {
        down.eb(LID[head[v]], LID[v]);
        v = parent[head[v]];
      } else {
        up.eb(LID[u], LID[head[u]]);
        u = parent[head[u]];
      }
    }
    if (LID[u] < LID[v]) down.eb(LID[u] + edge, LID[v]);
    elif (LID[v] + edge <= LID[u]) up.eb(LID[u], LID[v] + edge);
    reverse(all(down));
    up.insert(up.end(), all(down));
    return up;
  }

  vc<int> restore_path(int u, int v) {
    vc<int> P;
    for (auto &&[a, b]: get_path_decomposition(u, v, 0)) {
      if (a <= b) {
        FOR(i, a, b + 1) P.eb(V[i]);
      } else {
        FOR_R(i, b, a + 1) P.eb(V[i]);
      }
    }
    return P;
  }
};
#line 2 "alg/monoid/monoid_reverse.hpp"

template <class Monoid>
struct Monoid_Reverse {
  using value_type = typename Monoid::value_type;
  using X = value_type;
  static constexpr X op(const X &x, const X &y) { return Monoid::op(y, x); }
  static constexpr X unit() { return Monoid::unit(); }
  static const bool commute = Monoid::commute;
};
#line 6 "graph/ds/tree_monoid.hpp"

template <typename TREE, typename Monoid, bool edge>
struct Tree_Monoid {
  using MX = Monoid;
  using X = typename MX::value_type;
  TREE &tree;
  int N;
  SegTree<MX> seg;
  SegTree<Monoid_Reverse<MX>> seg_r;

  Tree_Monoid(TREE &tree) : tree(tree), N(tree.N) {
    build([](int i) -> X { return MX::unit(); });
  }

  Tree_Monoid(TREE &tree, vc<X> &dat) : tree(tree), N(tree.N) {
    build([&](int i) -> X { return dat[i]; });
  }

  template <typename F>
  Tree_Monoid(TREE &tree, F f) : tree(tree), N(tree.N) {
    build(f);
  }

  template <typename F>
  void build(F f) {
    if (!edge) {
      auto f_v = [&](int i) -> X { return f(tree.V[i]); };
      seg.build(N, f_v);
      if constexpr (!MX::commute) { seg_r.build(N, f_v); }
    } else {
      auto f_e = [&](int i) -> X {
        return (i == 0 ? MX::unit() : f(tree.v_to_e(tree.V[i])));
      };
      seg.build(N, f_e);
      if constexpr (!MX::commute) { seg_r.build(N, f_e); }
    }
  }

  void set(int i, X x) {
    if constexpr (edge) i = tree.e_to_v(i);
    i = tree.LID[i];
    seg.set(i, x);
    if constexpr (!MX::commute) seg_r.set(i, x);
  }

  void multiply(int i, X x) {
    if constexpr (edge) i = tree.e_to_v(i);
    i = tree.LID[i];
    seg.multiply(i, x);
    if constexpr (!MX::commute) seg_r.multiply(i, x);
  }

  X prod_path(int u, int v) {
    auto pd = tree.get_path_decomposition(u, v, edge);
    X val = MX::unit();
    for (auto &&[a, b]: pd) { val = MX::op(val, get_prod(a, b)); }
    return val;
  }

  // uv path 上で prod_path(u, x) が check を満たす最後の x

  // なければ （つまり path(u,u) が ng ）-1

  template <class F>
  int max_path(F check, int u, int v) {
    if constexpr (edge) return max_path_edge(check, u, v);
    if (!check(prod_path(u, u))) return -1;
    auto pd = tree.get_path_decomposition(u, v, edge);
    X val = MX::unit();
    for (auto &&[a, b]: pd) {
      X x = get_prod(a, b);
      if (check(MX::op(val, x))) {
        val = MX::op(val, x);
        u = (tree.V[b]);
        continue;
      }
      auto check_tmp = [&](X x) -> bool { return check(MX::op(val, x)); };
      if (a <= b) {
        // 下り

        auto i = seg.max_right(check_tmp, a);
        return (i == a ? u : tree.V[i - 1]);
      } else {
        // 上り

        int i = 0;
        if constexpr (MX::commute) i = seg.min_left(check_tmp, a + 1);
        if constexpr (!MX::commute) i = seg_r.min_left(check_tmp, a + 1);
        if (i == a + 1) return u;
        return tree.V[i];
      }
    }
    return v;
  }

  X prod_subtree(int u) {
    int l = tree.LID[u], r = tree.RID[u];
    return seg.prod(l + edge, r);
  }

  X prod_all() { return prod_subtree(tree.V[0]); }

  inline X get_prod(int a, int b) {
    if constexpr (MX::commute) {
      return (a <= b) ? seg.prod(a, b + 1) : seg.prod(b, a + 1);
    }
    return (a <= b) ? seg.prod(a, b + 1) : seg_r.prod(b, a + 1);
  }

private:
  template <class F>
  int max_path_edge(F check, int u, int v) {
    static_assert(edge);
    if (!check(MX::unit())) return -1;
    int lca = tree.lca(u, v);
    auto pd = tree.get_path_decomposition(u, lca, edge);
    X val = MX::unit();

    // climb

    for (auto &&[a, b]: pd) {
      assert(a >= b);
      X x = get_prod(a, b);
      if (check(MX::op(val, x))) {
        val = MX::op(val, x);
        u = (tree.parent[tree.V[b]]);
        continue;
      }
      auto check_tmp = [&](X x) -> bool { return check(MX::op(val, x)); };
      int i = 0;
      if constexpr (MX::commute) i = seg.min_left(check_tmp, a + 1);
      if constexpr (!MX::commute) i = seg_r.min_left(check_tmp, a + 1);
      if (i == a + 1) return u;
      return tree.parent[tree.V[i]];
    }
    // down

    pd = tree.get_path_decomposition(lca, v, edge);
    for (auto &&[a, b]: pd) {
      assert(a <= b);
      X x = get_prod(a, b);
      if (check(MX::op(val, x))) {
        val = MX::op(val, x);
        u = (tree.V[b]);
        continue;
      }
      auto check_tmp = [&](X x) -> bool { return check(MX::op(val, x)); };
      auto i = seg.max_right(check_tmp, a);
      return (i == a ? u : tree.V[i - 1]);
    }
    return v;
  }
};
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