Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices, whose i-th edge connects Vertex u_i and Vertex v_i.
Vertex i has an integer a_i written on it.
For every integer k from 1 through N, solve the following problem:

We will make a sequence by lining up the integers written on the vertices along the shortest path from Vertex 1 to Vertex k, in the order they appear. Find the length of the longest increasing subsequence of this sequence.

Here, the longest increasing subsequence of a sequence A of length L is the subsequence A_{i_1} , A_{i_2} , ... , A_{i_M} with the greatest possible value of M such that 1 \leq i_1 < i_2 < ... < i_M \leq L and A_{i_1} < A_{i_2} < ... < A_{i_M}.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq a_i \leq 10^9

1 \leq u_i , v_i \leq N

u_i \neq v_i

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N
u_1 v_1
u_2 v_2
:
u_{N-1} v_{N-1}

Output

Print N lines. The k-th line, print the length of the longest increasing subsequence of the sequence obtained from the shortest path from Vertex 1 to Vertex k.

Sample Input 1

10
1 2 5 3 4 6 7 3 2 4
1 2
2 3
3 4
4 5
3 6
6 7
1 8
8 9
9 10

Sample Output 1

1
2
3
3
4
4
5
2
2
3

For example, the sequence A obtained from the shortest path from Vertex 1 to Vertex 5 is 1,2,5,3,4. Its longest increasing subsequence is A_1, A_2, A_4, A_5, with the length of 4.

Solution in C++ (Clang 10.0.0) (judge verdict: Wrong Answer)

#include<iostream>
#include<cstdio>
#include<stack>
#include<queue>
#include<algorithm>
#include<string>
#include<set>
#include<map>
#include<vector>
#include<math.h>
#include<tuple>
#include<string.h>
#include <numeric> 
#include<unordered_map>
using namespace std; 
#define sim template < class c
#define ris return * this
#define dor > debug & operator <<
#define eni(x) sim > typename \
  enable_if<sizeof dud<c>(0) x 1, debug&>::type operator<<(c i) {
sim > struct rge { c b, e; };
sim > rge<c> range(c i, c j) { return rge<c>{i, j}; }
sim > auto dud(c* x) -> decltype(cerr << *x, 0);
sim > char dud(...);
struct debug {
#ifdef LOCAL
~debug() { cerr << endl; }
eni(!=) cerr << boolalpha << i; ris; }
eni(==) ris << range(begin(i), end(i)); }
sim, class b dor(pair < b, c > d) {
  ris << "(" << d.first << ", " << d.second << ")";
}
sim dor(rge<c> d) {
  *this << "[";
  for (auto it = d.b; it != d.e; ++it)
    *this << ", " + 2 * (it == d.b) << *it;
  ris << "]";
}
#else
sim dor(const c&) { ris; }
#endif
};
#define pp(...) "\033[94m [" << #__VA_ARGS__ ": " << (__VA_ARGS__) << "]\033[0m"
#define pp1(...) "\033[1;47;35m" << #__VA_ARGS__ ": " << (__VA_ARGS__) << "]\033[0m"
#define dg(x) debug() << pp1(x)
#define dg1(x) debug() << pp(x)
#define dg2(x,y) debug() << pp(x) pp(y)
#define dg3(x,y,z) debug() << pp(x) pp(y) pp(z)
#define dg4(w,x,y,z) debug() << pp(w) pp(x) pp(y) pp(z)
#define int long long
#define all(x) x.begin() , x.end()
#define clr(x) memset(x , 0 , sizeof x);
#define sz(x) (int)x.size()
template<typename T> void db(T x ){ debug() <<">>> " <<  x; }
template<typename T> void db(T x , T y){ debug() <<">>> " <<  x <<" , "<< y; }
template<typename T> void db(T x , T y , T z){ debug() <<">>> "<< x <<" , "<< y <<" , "<< z; }
template<typename T> void db(T x , T y , T z , T a){ debug() <<">>> "<< x <<" , " << y <<" , "<< z <<" , "<< a; }
const int xx = 2e5 + 10;
const int Mod = 1e9 + 7;
const int inf = 1e18;
const int Maxn = 20010;

vector<int> g[xx];
vector<int> ans , dp , A;
void dfs(int n , int p) {
  int id = lower_bound(all(dp) , A[n]) - dp.begin();
  int roll_back = dp[id];
  dp[id] = A[n];
  ans[n] = lower_bound(all(dp) , A[n]) - dp.begin();

  for(auto i : g[n]) {
    if(i != p) {
      dfs(i , n);
    }
  }
  dp[id] = roll_back;
}
signed main(){
  ios::sync_with_stdio(false);cin.tie(0);

  int N;
  cin >> N;
  ans = A = vector<int>(N);
  dp = vector<int>(N , inf);
  for(int i = 0; i < N; i++) {
    cin >> A[i];
  }
  for(int i = 0; i + 1 < N; i++) {
    int u , v;
    cin >> u >> v;
    u-- , v--;
    g[u].push_back(v);
    g[v].push_back(u);
  }
  dfs(0 , -1);

  for(int i : ans) {
    cout << i + 1 << "\n";
  }
}