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 = '\nimport math\nimport bisect\n\nn = int(input())\nalist = [0]+list(map(int,input().split()))\nuv_list = []\nfor i in range(n-1):\n    u,v = list(map(int,input().split()))\n    if u > v:\n        u, v = v, u\n    uv_list.append([u,v])\n\ntree = [[] for _ in range(n+1)]\nfor i in range(n-1):\n    uv = uv_list[i]\n    tree[uv[0]].append(uv[1])\n    tree[uv[1]].append(uv[0])\n    \n\nqueue = [1]\nused = [0]*(n+1)\nans = [0]*(n+1)\ndp = []\nstack = []\nwhile queue != []:\n    cur = queue[-1]\n\n    if used[cur] == 0:\n        if cur == 1:\n            dp.append(alist[cur])\n            stack.append([len(dp)-1,None])\n        else:\n            if alist[cur] > dp[-1]:\n                dp.append(alist[cur])\n                stack.append([len(dp)-1,None])\n            else:\n                offset = bisect.bisect_left(dp,alist[cur])\n                stack.append([offset,dp[offset]])\n                dp[offset] = alist[cur]\n        ans[cur] = len(dp)\n\n    if tree[cur] != [] and used[cur] == 0:\n        for child in tree[cur]:\n            if used[child] == 0:\n                queue.append(child)\n    else:\n        queue.pop()\n\n        #reverse\n        back = stack.pop()\n        if back[1] == None:\n            dp.pop()\n        else:\n            dp[back[0]] = back[1]\n    used[cur] = 1\nfor i in range(n):\n    print((ans[i+1]))\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 14323