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 = 'import sys\nimport math\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_HALF_EVEN\nfrom collections import deque\ndef I(): return int(sys.stdin.readline())\ndef MI(): return map(int, sys.stdin.readline().split())\ndef LI(): return list(map(int, sys.stdin.readline().split()))\ndef LI2(N): return [list(map(int, sys.stdin.readline().split())) for i in range(N)]\ndef S(): return sys.stdin.readline().rstrip()\ndef LS(): return sys.stdin.readline().split()\ndef LS2(N): return [sys.stdin.readline().split() for i in range(N)]\ndef FILL(i,h): return [i for j in range(h)]\ndef FILL2(i,h,w): return [FILL(i,w) for j in range(h)]\ndef FILL3(i,h,w,d): return [FILL2(i,w,d) for j in range(h)]\ndef FILL4(i,h,w,d,d2): return [FILL3(i,w,d,d2) for j in range(h)]\ndef sisha(num,digit): return Decimal(str(num)).quantize(Decimal(digit),rounding=ROUND_HALF_UP)\n#\'0.01\'や\'1E1\'などで指定、整数に戻すならintをかます\nMOD = 1000000007\nINF = float("inf")\nsys.setrecursionlimit(10**6+10)\n#input = sys.stdin.readline\nfrom bisect import bisect_left\n\ndef dfs(i,before):\n    nonlocal seq\n    nonlocal ans\n    added  = 0\n\n    #現在地のAの値を、以前までのseqリストのどこに追加するか決める\n    pos = bisect_left(seq,a[i-1])\n    old = seq[pos]\n    seq[pos]=a[i-1]\n    ans[i-1]=bisect_left(seq,INF)\n\n    #隣接する頂点に関して再帰\n    for u in to[i]:\n        if u==before:\n            continue\n        dfs(u,i)\n\n    #seq配列をもとに戻す\n    seq[pos]=old\n\n\nN = I()\na = LI()\nto = [[] for i in range(N+1)]\nto[0] += [1]\nfor i in range(N-1):\n    u,v = MI()\n    to[u].append(v)\n    to[v].append(u)\nseq = [INF]*N\nans = [-1]*N\n\ndfs(1,-1)\n[print(i) for i in ans]\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: