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 = '#!/usr/bin/env python3\nimport sys\nsys.setrecursionlimit(10**8)\nfrom bisect import bisect_left\nINF = float("inf")\n\n\nfrom collections import defaultdict\n\n\nclass Graph(object):\n    def __init__(self, N):\n        self.N = N\n        self.E = defaultdict(list)\n\n    def add_edge(self, f, t, w=1):\n        self.E[f].append((t, w))\n        self.E[t].append((f, w))\n\n\ndef solve(N: int, a: "List[int]", u: "List[int]", v: "List[int]"):\n\n    g = Graph(N)\n    for x, y in zip(u, v):\n        g.add_edge(x-1, y-1)\n\n    ans = [0]*N\n\n    dp = [INF]*N\n    event = []\n    var = {"last": 0}\n\n    def dfs(curr, par):\n        i = bisect_left(dp, a[curr])\n        if dp[i] > a[curr]:\n            if dp[i] == INF:\n                var["last"] = i\n            event.append((curr, i, dp[i]))\n            dp[i] = a[curr]\n\n        ans[curr] = var["last"]+1\n\n        for child, w in g.E[curr]:\n            if child == par:\n                continue\n            dfs(child, curr)\n\n        c, i, v = event[-1]\n        if c == curr:\n            dp[i] = v\n            if v == INF:\n                var["last"] = i-1\n            event.pop()\n\n    dfs(0, -1)\n    print(*ans, sep="\\n")\n\n    return\n\n\ndef main():\n\n    def iterate_tokens():\n        for line in sys.stdin:\n            for word in line.split():\n                yield word\n    tokens = iterate_tokens()\n    N = int(next(tokens))  # type: int\n    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"\n    u = [int()] * (N - 1)  # type: "List[int]"\n    v = [int()] * (N - 1)  # type: "List[int]"\n    for i in range(N - 1):\n        u[i] = int(next(tokens))\n        v[i] = int(next(tokens))\n    solve(N, a, u, v)\n\n\ndef __starting_point():\n    main()\n\n__starting_point()'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 18553