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
import java.util.*;
import java.io.*;
import static java.lang.Math.*;

public class Main {
    
    static int[] a;
	public static void main(String[] args) throws Exception {
		Scanner sc = new Scanner(System.in);
		
		// 入力
		int n = sc.nextInt();
		a = new int[n];
		for(int i = 0; i < n; i++){
		    a[i] = sc.nextInt();
		}
		Tree tr = new Tree(n, false);
		for(int i = 0; i < n-1; i++){
		    int from = sc.nextInt()-1;
		    int to = sc.nextInt()-1;
		    tr.add(from, to);
		}
		
		// 計算
		//int result = 0;
		long[] result = tr.dfs(0);
		
		// 出力
		for(int i = 0; i < n; i++){
		    System.out.println(result[i]);
		}
		//System.out.println(result);
	}
	
	public static class Scanner {
		private BufferedReader br;
		private StringTokenizer tok;

		public Scanner(InputStream is) throws IOException {
			br = new BufferedReader(new InputStreamReader(is));
		}
		private void getLine() throws IOException {
			while(!hasNext()){
				tok = new StringTokenizer(br.readLine());
			}
		}
		private boolean hasNext() {
			return tok != null && tok.hasMoreTokens();
		}
		public String next() throws IOException {
			getLine();
			return tok.nextToken();
		}
		public int nextInt() throws IOException {
			return Integer.parseInt(next());
		}
		public long nextLong() throws IOException {
			return Long.parseLong(next());
		}
		public double nextDouble() throws IOException {
			return Double.parseDouble(next());
		}
	}
}

class Tree {
	int n;
	boolean directed;
	ArrayList<Edge>[] adjList;
	int m;
	long[] dist;
	int start;
	int[] prev;
	ArrayList<Integer> path;
	long INF = 1001001001001001001L;

	public Tree (int n, boolean directed) {
		this.n = n;
		this.m = 0;
		this.directed = directed;
		this.adjList = new ArrayList[n];
		for(int i = 0; i < n; i++){
			adjList[i] = new ArrayList<Edge>();
		}
	}
	
	public void add(int from, int to) {
		add(from, to, 1);
	}
	public void add(int from, int to, int w) {
		adjList[from].add(new Edge(from, to, w));
		this.m++;
		if(!directed){
			adjList[to].add(new Edge(to, from, w));
			this.m++;
		}
	}
	
	public long[] dfs(int start) { // O(M)
		this.start = start;
		this.prev = new int[n];
		Arrays.fill(prev, -1);
		prev[start] = start;
		this.dist = new long[n];
		Arrays.fill(dist, INF);
		dist[start] = 0;
		
		long[] ans = new long[n];
		int[] list = new int[n];
		Arrays.fill(list, Integer.MAX_VALUE);
		list[0] = Main.a[start];
		ans[0] = 1;
		dfsFunc(start, list, ans, 1);
		
		return ans;
	}
	
	private void dfsFunc(int now, int[] list, long[] ans, int len) {
	    for(Edge e : adjList[now]){
	        if(dist[e.to] != INF) continue;
			dist[e.to] = dist[now] + e.w;
			prev[e.to] = now;
			
			int index = binarySearch(list, Main.a[e.to], true);
			int old = list[index];
			list[index] = Main.a[e.to];
			ans[e.to] = max(len, index+1);
			dfsFunc(e.to, list, ans, (int)ans[e.to]);
			list[index] = old;
		}
	}

	public int binarySearch(int[] array, int n, boolean isStrict){
		int index = 0;
		int l = -1;
		int r = array.length;
		while(r-l > 1){
			int c = (r + l)/2;
			
			if(n < array[c] || (isStrict && n == array[c])){
				r = c;
			}else{
				l = c;
			}
		}
		index = r;
		return index;
	}
	
	public ArrayList<Integer> pathTo(int to){
		this.path = new ArrayList<Integer>();
		pathToRec(to);
		return path;
	}
	
	private void pathToRec(int to){
		if(to != start) pathToRec(prev[to]);
		path.add(to);
		return;
	}
	
	public String toString(){
		StringBuilder sb = new StringBuilder();
		for(ArrayList<Edge> list : adjList){
			for(Edge e : list){
				sb.append(e.toString());
				sb.append("\n");
			}
		}
		return sb.toString();
	}
	
	class Edge {
		int from;
		int to;
		int w;
		
		Edge(int from, int to, int w){
			this.from = from;
			this.to = to;
			this.w = w;
		}
		
		public String toString(){
			return "from:" + from + " to:" + to + " w:" + w;
		}
	}
	
	class Vertex {
		int id;
		long distance;
		
		Vertex(int id, long distance){
			this.id = id;
			this.distance = distance;
		}
		
		public String toString(){
			return "id:" + id + " distance:" + distance;
		}
	}
}


