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\nsys.setrecursionlimit(2147483647)\n\nn=int(input())\na=list(map(int,input().split()))\n\ntree=[[] for _ in range(n)]\nfor _ in range(n-1):\n    aa,bb=list(map(int,input().split()))\n    tree[aa-1].append(bb-1)\n    tree[bb-1].append(aa-1)\n\nans=[0]*n\nimport bisect\n\n#bisect.bisect_left(a, 4) #被ったら左側のindexを返す\n#bisect.bisect_right(a, 4)#被ったら右側のindexを返す\n\ndef lisOnTree(x,lis):\n    nonlocal ans,tree,a,n\n    i=bisect.bisect_left(lis, a[x])\n    if i==len(lis):\n        lis.append(a[x])\n        flag="append"\n    else:\n        flag=(i,lis[i])\n        lis[i]=a[x]\n    #print(x,lis)\n    ans[x]=len(lis)\n    for item in tree[x]:\n        if ans[item]==0:\n            lisOnTree(item,lis)\n    if flag=="append":\n        lis.pop()\n    else:\n        lis[flag[0]]=flag[1]\n\nlisOnTree(0,[])\n\nfor item in ans:\n    print(item)\n\n\n    \n\n\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: