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 = "# 木によってLISを作り、dfsで抜けるときにLISをその前の状態まで復元する\n\nimport sys\nsys.setrecursionlimit(1 << 25)\nread = sys.stdin.readline\nra = range\nenu = enumerate\n\n\ndef mina1(x):\n    return int(x) - 1\n\n\ndef read_ints(mina=None):\n    ret = map(int, read().split())\n    if mina == None:\n        return list(ret)\n    else:\n        return list(map(lambda x: x - mina, ret))\n\n\ndef read_a_int():\n    return int(read())\n\n\nfrom bisect import bisect_left, bisect_right\nfrom collections import defaultdict\nN = read_a_int()\nA = read_ints()\ntree = defaultdict(lambda: [])\nfor _ in ra(N - 1):\n    u, v = read_ints(mina=1)\n    tree[u].append(v)\n    tree[v].append(u)\n\nLIS = []\nans = [0] * N  # 各ノードのlen(LIS)を記録\n\n\ndef dfs(now, p):  # 現在のノード、親\n    a = A[now]\n    # LISの更新\n    idx = bisect_left(LIS, a)\n    is_append = False\n    if idx == len(LIS):\n        LIS.append(a)\n        is_append = True\n    else:\n        old = LIS[idx]   # なんの値だったか持っておく\n        LIS[idx] = a  # aに更新\n\n    ans[now] = len(LIS)  # 答えを記録\n    # 次のノードを探索\n    for to in tree[now]:\n        if to == p:\n            continue\n        dfs(to, now)\n\n    # 抜けるときにLISを復元\n    if is_append:\n        del LIS[idx]\n    else:\n        LIS[idx] = old\n\n\ndfs(0, -1)\nprint(*ans, sep='\\n')\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 14419