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1、题目大意:就是在动态的树上路径权值第k大。
2、分析:这个就是树链剖分+树套树
3、代码:

#include <cstdio> 
#include <cstdlib>
#include <cstring>
#include <algorithm>
using namespace std;
#define M 1000000
int Height[M], Top[M], value[M], num[M], Size[M], Fa[M];
int ST_tot, tot;
int son[M], head[M], Next[M];
int n, m;
struct Node{
    Node *ch[2];
    int cnt, num, r, v;
    bool operator < (const Node& rhs) const{
        return r < rhs.r;
    }
    int cmp(int x){
        if(x == v) return -1;
        if(x < v) return 0;
        return 1;
    }
    void maintain(){
        cnt = num;
        if(ch[0]) cnt += ch[0] -> cnt;
        if(ch[1]) cnt += ch[1] -> cnt;
    }
} *root[2 * M], ft[5 * M];
int treap_tot;
inline void treap_rotate(Node* &o, int d){
    Node* k = o -> ch[d ^ 1];
    o -> ch[d ^ 1] = k -> ch[d];
    k -> ch[d] = o;
    o -> maintain();
    k -> maintain();
    o = k;
    return;
}
inline void treap_insert(Node* &o, int x){
    if(o == NULL){
        o = &ft[treap_tot ++];
        o -> ch[0] = o -> ch[1] = NULL;
        o -> cnt = o -> num = 1;
        o -> v = x;
        o -> r = rand();
    }
    else{
        int d = o -> cmp(x);
        if(d == -1){
            o -> num ++;
        }
        else{
            treap_insert(o -> ch[d], x);
            if(o < o -> ch[d]) treap_rotate(o, d ^ 1);
        }
    }
    o -> maintain();
}
inline void treap_remove(Node* &o, int x){
    int d = o -> cmp(x); 
    if(d == -1){
        if(o -> num > 1) o -> num --;
        else if(o -> ch[0] == NULL) o = o -> ch[1];
        else if(o -> ch[1] == NULL) o = o -> ch[0];
        else {
            int d2;
            if(o -> ch[0] > o -> ch[1]) d2 = 1;
            else d2 = 0;
            treap_rotate(o, d2); 
            treap_remove(o -> ch[d2], x);
        }
    }
    else treap_remove(o -> ch[d], x);
    if(o) o -> maintain();
}
inline int treap_lessk(Node* &o, int k){
    if(o == NULL) return 0; 
    int d = o -> cmp(k);
    if(d == -1){
        int ret = 0;
        if(o -> ch[0]) ret += o -> ch[0] -> cnt;
        return ret;
    }
    else if(d == 0){
        return treap_lessk(o -> ch[0], k);
    }
    else{
        int ss = o -> num;
        if(o -> ch[0]) ss += o -> ch[0] -> cnt;
        return treap_lessk(o -> ch[1], k) + ss;
    }
}
inline void init(){
    Top[1] = 1;
    memset(head, -1, sizeof(head));
    tot = ST_tot = 0;
}
inline void add(int l, int r, int o, int x, int y, int z){
    if(y != -1) treap_remove(root[o], y);
    treap_insert(root[o], z);
    if(l == r) {
        return;
    }
    int mid = (l + r) / 2;
    if(x <= mid) add(l, mid, 2 * o, x, y, z);
    else add(mid + 1, r, 2 * o + 1, x, y, z);
}
inline int query(int l, int r, int o, int x, int y, int z){
    if(x <= l && r <= y) return root[o] -> cnt - treap_lessk(root[o], z);
    int ret = 0, mid = (l + r) / 2;
    if(x <= mid) ret += query(l, mid, 2 * o, x, y, z);
    if(y > mid) ret += query(mid + 1, r, 2 * o + 1, x, y, z);
    return ret;
}
inline void insert(int x, int y){
    tot ++;
    son[tot] = y;
    Next[tot] = head[x];
    head[x] = tot;
}
inline void dfs1(int x, int fa, int height){
    Fa[x] = fa;
    Height[x] = height;
    Size[x] = 1;
    for(int i = head[x]; i != -1; i = Next[i]) if(son[i] != fa){
        dfs1(son[i], x, height + 1);
        Size[x] += Size[son[i]];
    }
}
inline void dfs2(int x, int fa){
    ++ ST_tot;
    num[x] = ST_tot;
    add(1, n, 1, ST_tot, -1, value[x]);
    int o = 0, ss = 0;
    for(int i = head[x]; i != -1; i = Next[i]) if(son[i] != fa){
        if(Size[son[i]] > ss){
            ss = Size[son[i]];
            o = i;
        }
    }
    if(o != 0){
        Top[son[o]] = Top[x];
        dfs2(son[o], x);
    }
    for(int i = head[x]; i != -1; i = Next[i]) if(son[i] != fa && o != i){
        Top[son[i]] = son[i];
        dfs2(son[i], x);
    }
}
inline void real_add(int x, int y){
    add(1, n, 1, num[x], value[x], y);
    value[x] = y;
}
inline int check(int x, int y, int k){
    int ret = 0;
    while(Top[x] != Top[y]){
        if(Height[Top[x]] < Height[Top[y]]) swap(x, y);
        ret += query(1, n, 1, num[Top[x]], num[x], k);
        x = Fa[Top[x]];
    }
    if(Height[x] < Height[y]) swap(x, y);
    ret += query(1, n, 1, num[y], num[x], k);
    return ret;
}
inline int real_query(int x, int y, int k){
    int l = -1, r = 100000000;
    while(l < r){
        int mid = (l + r) / 2;
        if(mid == l) mid ++;
        if(check(x, y, mid) >= k) l = mid;
        else r = mid - 1;  
    }
    if(l == -1) return -1;
    return l;
}
int main(){
    scanf("%d%d", &n, &m);
    init();
    for(int i = 1; i <= n; i ++) scanf("%d", &value[i]);
    for(int i = 1; i < n; i ++){
        int x, y;
        scanf("%d%d", &x, &y);
        insert(x, y);
        insert(y, x);
    }
    dfs1(1, 0, 1);
    dfs2(1, 0);
    for(int i = 1; i <= m; i ++){
        int k, x, y;
        scanf("%d%d%d", &k, &x, &y);
        if(k == 0){
            real_add(x, y);
        }
        else if(k > 0){
            int qq = real_query(x, y, k);
            if(qq == -1){
                printf("invalid request!\n");
            }
            else{
                printf("%d\n", qq);
            }
        }
    }
    return 0;
}