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.

Example

Input

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


Output

1
2
3
3
4
4
5
2
2
3
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef vector<ll> vec;
typedef vector<vec> mat;
typedef pair<ll,ll> pll;
const ll mod=1e9+7;
//const ll mod=998244353;
const ll inf=5e18;

vec ans(2e5+10),dp(2e5+10),a(2e5+10);
ll n;
vector<set<ll>> G(2e5+10);

void dfs(ll f, ll ff,ll R) {
  ll l=0,r=R;
  while(r-l>1) {
    ll m=(r+l)/2;
    if(dp[m]<a[f]) l=m;
    else r=m;
  }
  ll pre=dp[r];
  dp[r]=a[f];
  if(r==R) ans[f]=R++;
  else ans[f]=R-1;
  for(ll t:G[f]) if(t!=ff) dfs(t,f,R);
  dp[r]=pre;
}

int main() {
  cin >> n;
  for(ll i=0;i<n;i++) cin >> a[i];
  for(ll i=0;i<n-1;i++) {
    ll u,v;
    cin >> u >> v;
    u--,v--;
    G[u].insert(v);
    G[v].insert(u);
  }
  for(ll i=1;i<n;i++) dp[i]=inf;
  dfs(0,-1,1);
  for(ll i=0;i<n;i++) cout << ans[i] << endl;
}