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-----
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
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.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

ns = {"__name__": "__main__"}
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = '## まきもどし？\nma = lambda :map(int,input().split())\nlma = lambda :list(map(int,input().split()))\ntma = lambda :tuple(map(int,input().split()))\nni = lambda:int(input())\nyn = lambda fl:print("Yes") if fl else print("No")\nimport collections\nimport math\nimport itertools\nimport heapq as hq\nceil = math.ceil\nimport sys\nsys.setrecursionlimit(10**6)\nn = ni()\nA = lma()\ntree = [[] for i in range(n)]\nINF = 10**10\nfor i in range(n-1):\n    u,v = ma();u-=1;v-=1\n    tree[u].append(v)\n    tree[v].append(u)\n#print(tree)\ndef isok(num,val):##適宜変更\n    return num<val\ndef bisect(ls,val): ##valの関数isok(x,val)がTrueとなる一番右のindex を返す 全部Falseなら-1,Trueならlen(ls)-1\n    ok = -1\n    ng = len(ls)\n    x = (ok+ng)//2\n    while ng-ok>1:\n        num = ls[x]\n        if isok(num,val):\n            ok = x\n        else:\n            ng = x\n        x = (ok+ng)//2\n    return ok ##一番右のTrueのindex  Trueの個数はok+1こ\ndef LIS_1(x,ls,right):##right ::これまでわかっている右端\n    idx = bisect(ls,x)+1\n    if idx+1>right:\n        ret= idx+1\n    else:\n        ret=right\n    rewind_idxval.append((idx,ls[idx]))\n    ls[idx]=x\n    #print(x,ls)\n    return ret\ndef DFS_LIS(prev,ls,right):\n    ans[prev]= LIS_1(A[prev],ls,right)\n    for node in tree[prev]:\n        if not visited[node]:\n            visited[node]=True\n            DFS_LIS(node,ls,ans[prev])\n    #rewind\n    idx,pval = rewind_idxval.pop()\n    ls[idx] = pval\n\nans = [0]*n\ns = 0\nright=0\nrewind_idxval=collections.deque()\nvisited=[False]*n\nvisited[s]=True\nls=[INF]*n\nDFS_LIS(s,ls,right)\n#print(ls)\nfor a in ans :\n    print(a)\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 14255