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 = "# coding: utf-8\nimport sys\n#from operator import itemgetter\nsysread = sys.stdin.readline\n#from heapq import heappop, heappush\n#from collections import defaultdict\nsys.setrecursionlimit(10**7)\n#import math\n#from itertools import combinations\ndef run():\n    N = int(input())\n    A = [0] + list(map(int, sysread().split()))\n    to = [[] for _ in range(N+1)]\n    for _ in range(N-1):\n        u,v = map(int, sysread().split())\n        to[u].append(v)\n        to[v].append(u)\n    seen = [False] * (N+1)\n    dp = [float('inf')] * (N+2)\n    dp[0] = -float('inf')\n    ddp2 = [0]\n    paths = []\n    actions = []# (idx, pre, pro)\n    ans = [0] * (N+1)\n    def dfs(node, parent=None):\n\n        a = A[node]\n        seen[node] = True\n        if parent == None:\n            actions.append((1, dp[1], a, 1))\n            dp[1] = a\n            ddp2[0] += 1\n        else:\n            idx = bin_search(dp, a)\n            if dp[idx] == float('inf'):\n                actions.append((idx, dp[idx], a, 1))\n                ddp2[0] += 1\n            else:\n                actions.append((idx, dp[idx], a, 0))\n            dp[idx] = a\n        ans[node] = ddp2[0]\n        for next in to[node]:\n            if not seen[next]:\n                dfs(next, node)\n\n        idx, pre, pro, change = actions.pop()\n        dp[idx] = pre\n        ddp2[0] -= change\n\n    dfs(1)\n    for s in ans[1:]:\n        print(s)\n    return None\n\ndef bin_search(A, x):\n    '''\n    return index which is lowest in values more than or equal to x\n    '''\n    low = 0\n    high = len(A) - 1\n    c = (low + high) // 2\n    if A[-1] < x: return float('inf')\n    while high - low > 1:\n        if A[c] < x:\n            low = c\n            c = (low + high) // 2\n\n        elif A[c] > x:\n            high = c\n            c = (high + low) // 2\n        else:\n            return c\n    return high\n\ndef __starting_point():\n    run()\n__starting_point()"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 14917