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 numpy as np\n#import math\n#from decimal import *\n#from numba import njit\nfrom bisect import bisect_left\n\n#@njit\ndef main():\n    N = int(input())\n    a = list(map(int, input().split()))\n    edges = [[] for _ in range(N+1)]\n    for _ in range(N-1):\n        u,v = list(map(int, input().split()))\n        edges[u] += v,\n        edges[v] += u,\n\n    ans = [0]*(N+1)\n    lastLis = []\n    visited = [False]*(N+1)\n    parent = {}\n    n = 1\n    stack = []\n    while True:\n        #print(n, lastLis, stack)\n        if not visited[n]:\n            if len(lastLis) == 0 or a[n-1] > lastLis[-1]:\n                lastLis.append(a[n-1])\n                stack += (len(lastLis)-1, -1),\n            else:\n                index = bisect_left(lastLis, a[n-1])\n                stack += (index, lastLis[index]),\n                lastLis[index] = a[n-1]\n            ans[n] = len(lastLis)\n            visited[n] = True\n        hasNext = False\n        for d in edges[n]:\n            if not visited[d]:\n                parent[d] = n\n                n = d\n                hasNext = True\n        if not hasNext:\n            if n == 1:\n                n = 0\n                break\n            n = parent[n]\n            # 巻き戻し\n            index, value = stack.pop(len(stack)-1)\n            if value < 0:\n                lastLis.pop(index)\n            else:\n                lastLis[index] = value\n\n    for i in range(1, len(ans)):\n        print((ans[i]))\n\n\n\n\nmain()\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 14471