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\nimport heapq\nimport bisect\nsys.setrecursionlimit(10**7)\nINTMAX = 9223372036854775807\nINTMIN = -9223372036854775808\nDVSR = 1000000007\ndef POW(x, y): return pow(x, y, DVSR)\ndef INV(x, m=DVSR): return pow(x, m - 2, m)\ndef DIV(x, y, m=DVSR): return (x * INV(y, m)) % m\ndef LI(): return map(int, sys.stdin.readline().split())\ndef LF(): return [float(x) for x in sys.stdin.readline().split()]\ndef LS(): return sys.stdin.readline().split()\ndef II(): return int(sys.stdin.readline())\n\n\ndef FLIST(n):\n    res = [1]\n    for i in range(1, n+1):\n        res.append(res[i-1]*i % DVSR)\n    return res\n\n\nN = II()\nAS = list(LI())\n\nMP = {}\n\nMX = 10**10\nLIS = [10**10]*N\nRES = [1]*N\nfor i in range(N-1):\n    a, b = LI()\n    a -= 1\n    b -= 1\n    if not a in MP:\n        MP[a] = []\n    if not b in MP:\n        MP[b] = []\n    MP[a].append(b)\n    MP[b].append(a)\n\n\ndef dfs(v, mp, lis, p):\n    for u in mp[v]:\n        if u != p:\n            i = bisect.bisect_left(lis, AS[u])\n            bef = lis[i]\n            lis[i] = AS[u]\n            longest = bisect.bisect_left(lis, MX)\n            RES[u] = longest\n            dfs(u, mp, lis, v)\n            lis[i] = bef\n\n\nLIS[0] = AS[0]\ndfs(0, MP, LIS, -1)\n\nprint(*RES, sep='\\n')\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 14785