题目链接：Count on a tree
大概可以认为是树上主席树的板子
我在之前的某些题解提到了，主席树一般来说有两个基本功能：
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可持久化功能，可以选择回退或者新增版本。
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对于可差性问题，可以有更好的转化为前缀和做法，常见的问题为权值类型问题。
在树上的路径第 \(k\) 大，显然如果我们能拿到这条路径上的权值树显然就随便做二分了，很基础的问题。
树上前缀和：
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关于点的树上前缀和，我们对于一条路径上的点假如设：\(pre[x]\) 为从 \(1 \sim x\) 的所有点，那么有 \(u \rightarrow v\) 上的所有点可以抽象的表示为：\(pre[u]+pre[v]-pre[lca]-pre[fa[lca]]\)
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关于边的树上前缀和，\(pre[x]\) 表示 \(1 \sim x\) 的所有边，那么 \(u \rightarrow v\) 的所有边可以表示为：\(pre[u]+pre[v]-2\times pre[lca]\)
如图所示，这也是树上前缀和问题的常见两种模型。本题显然是常见的点模型，那么树上前缀和就是一个很典型的差分型问题，那么我们就可以通过四棵权值树同时做贡献即可拿到正确的权值树。同时初始化，根据树上前缀和初始化规则，直接在 \(dfs\) 时我们做树上前缀和更新即可。最后，我们需要注意离散化值域以及求 \(lca\)，这里就使用倍增求了。
参照代码
#include <bits/stdc++.h>
// #pragma GCC optimize(2)
// #pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
// #pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")
#define isPbdsFile
#ifdef isPbdsFile
#include <bits/extc++.h>
#else
#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/trie_policy.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/list_update_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/exception.hpp>
#include <ext/rope>
#endif
using namespace std;
using namespace __gnu_cxx;
using namespace __gnu_pbds;
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef tuple<int, int, int> tii;
typedef tuple<ll, ll, ll> tll;
typedef unsigned int ui;
typedef unsigned long long ull;
typedef __int128 i128;
#define hash1 unordered_map
#define hash2 gp_hash_table
#define hash3 cc_hash_table
#define stdHeap std::priority_queue
#define pbdsHeap __gnu_pbds::priority_queue
#define sortArr(a, n) sort(a+1,a+n+1)
#define all(v) v.begin(),v.end()
#define yes cout<<"YES"
#define no cout<<"NO"
#define Spider ios_base::sync_with_stdio(false);cin.tie(nullptr);cout.tie(nullptr);
#define MyFile freopen("..\\input.txt", "r", stdin),freopen("..\\output.txt", "w", stdout);
#define forn(i, a, b) for(int i = a; i <= b; i++)
#define forv(i, a, b) for(int i=a;i>=b;i--)
#define ls(x) (x<<1)
#define rs(x) (x<<1|1)
#define endl '\n'
//用于Miller-Rabin
[[maybe_unused]] static int Prime_Number[13] = {0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};
template <typename T>
int disc(T* a, int n)
{
return unique(a + 1, a + n + 1) - (a + 1);
}
template <typename T>
T lowBit(T x)
{
return x & -x;
}
template <typename T>
T Rand(T l, T r)
{
static mt19937 Rand(time(nullptr));
uniform_int_distribution<T> dis(l, r);
return dis(Rand);
}
template <typename T1, typename T2>
T1 modt(T1 a, T2 b)
{
return (a % b + b) % b;
}
template <typename T1, typename T2, typename T3>
T1 qPow(T1 a, T2 b, T3 c)
{
a %= c;
T1 ans = 1;
for (; b; b >>= 1, (a *= a) %= c)if (b & 1)(ans *= a) %= c;
return modt(ans, c);
}
template <typename T>
void read(T& x)
{
x = 0;
T sign = 1;
char ch = getchar();
while (!isdigit(ch))
{
if (ch == '-')sign = -1;
ch = getchar();
}
while (isdigit(ch))
{
x = (x << 3) + (x << 1) + (ch ^ 48);
ch = getchar();
}
x *= sign;
}
template <typename T, typename... U>
void read(T& x, U&... y)
{
read(x);
read(y...);
}
template <typename T>
void write(T x)
{
if (typeid(x) == typeid(char))return;
if (x < 0)x = -x, putchar('-');
if (x > 9)write(x / 10);
putchar(x % 10 ^ 48);
}
template <typename C, typename T, typename... U>
void write(C c, T x, U... y)
{
write(x), putchar(c);
write(c, y...);
}
template <typename T11, typename T22, typename T33>
struct T3
{
T11 one;
T22 tow;
T33 three;
bool operator<(const T3 other) const
{
if (one == other.one)
{
if (tow == other.tow)return three < other.three;
return tow < other.tow;
}
return one < other.one;
}
T3() { one = tow = three = 0; }
T3(T11 one, T22 tow, T33 three) : one(one), tow(tow), three(three)
{
}
};
template <typename T1, typename T2>
void uMax(T1& x, T2 y)
{
if (x < y)x = y;
}
template <typename T1, typename T2>
void uMin(T1& x, T2 y)
{
if (x > y)x = y;
}
struct Hash
{
static uint64_t splitmix64(uint64_t x)
{
x += 0x9e3779b97f4a7c15;
x = (x ^ x >> 30) * 0xbf58476d1ce4e5b9;
x = (x ^ x >> 27) * 0x94d049bb133111eb;
return x ^ x >> 31;
}
static size_t get(const uint64_t x)
{
static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
return splitmix64(x + FIXED_RANDOM);
}
template <typename T>
size_t operator()(T x) const
{
return get(std::hash<T>()(x));
}
template <typename F, typename S>
size_t operator()(pair<F, S> p) const
{
return get(std::hash<F>()(p.first)) ^ std::hash<S>()(p.second);
}
};
constexpr int N = 1e5 + 10;
constexpr int T = ceil(log2(N));
int a[N], ord[N];
hash2<int, int, Hash> mp;
int fa[N][T + 1], deep[N];
vector<int> child[N];
int n, m, mx;
struct Node
{
int left, right, cnt;
} node[N << 5];
int root[N];
#define left(x) node[x].left
#define right(x) node[x].right
#define cnt(x) node[x].cnt
int cnt;
inline void add(const int pre, int& curr, const int val, const int l = 1, const int r = mx)
{
node[curr = ++cnt] = node[pre];
cnt(curr)++;
const int mid = l + r >> 1;
if (l == r)return;
if (val <= mid)add(left(pre),left(curr), val, l, mid);
else add(right(pre),right(curr), val, mid + 1, r);
}
inline int query(const int u, const int v, const int lca, const int lca_fa, const int k, const int l = 1,
const int r = mx)
{
if (l == r)return ord[l];
const int leftSize = cnt(left(u)) + cnt(left(v)) - cnt(left(lca)) - cnt(left(lca_fa));
const int mid = l + r >> 1;
if (leftSize >= k)return query(left(u),left(v),left(lca),left(lca_fa), k, l, mid);
return query(right(u),right(v),right(lca),right(lca_fa), k - leftSize, mid + 1, r);
}
inline void dfs(const int curr, const int pa)
{
deep[curr] = deep[fa[curr][0] = pa] + 1;
add(root[pa], root[curr], a[curr]);
forn(i, 1, T)fa[curr][i] = fa[fa[curr][i - 1]][i - 1];
for (const int nxt : child[curr])if (nxt != pa)dfs(nxt, curr);
}
inline int LCA(int x, int y)
{
if (deep[x] < deep[y])swap(x, y);
forv(i, T, 0)if (deep[fa[x][i]] >= deep[y])x = fa[x][i];
if (x == y)return x;
forv(i, T, 0)if (fa[x][i] != fa[y][i])x = fa[x][i], y = fa[y][i];
return fa[x][0];
}
int last;
inline void solve()
{
cin >> n >> m;
forn(i, 1, n)cin >> a[i], ord[i] = a[i];
sortArr(ord, n), mx = disc(ord, n);
forn(i, 1, mx)mp[ord[i]] = i;
forn(i, 1, n)a[i] = mp[a[i]];
forn(i, 1, n-1)
{
int u, v;
cin >> u >> v;
child[u].push_back(v), child[v].push_back(u);
}
dfs(1, 0);
while (m--)
{
int u, v, k;
cin >> u >> v >> k, u ^= last;
const int lca = LCA(u, v);
const int lca_fa = fa[lca][0];
cout << (last = query(root[u], root[v], root[lca], root[lca_fa], k)) << endl;
}
}
signed int main()
{
// MyFile
Spider
//------------------------------------------------------
// clock_t start = clock();
int test = 1;
// read(test);
// cin >> test;
forn(i, 1, test)solve();
// while (cin >> n, n)solve();
// while (cin >> test)solve();
// clock_t end = clock();
// cerr << "time = " << double(end - start) / CLOCKS_PER_SEC << "s" << endl;
}