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 python\n# -*- coding: utf-8 -*-\n\nimport sys\nsys.setrecursionlimit(10**7)\nfrom pprint import pprint as pp\nfrom pprint import pformat as pf\n\nimport math\nimport bisect\n\nclass Tree:\n    """\n    node id starts from 1\n    """\n\n    DUMMY = 0 # used as dummy_node_id and dummy_node_value\n\n    def __init__(self, num_node, node_values):\n        self.node_values = [-1] + node_values # [0] is dummy\n        self.edges = [None] * (num_node + 1) # [0] is dummy, var[a] = set of links\n        for i, _ in enumerate(self.edges):\n            self.edges[i] = set()\n        self.seq = [self.DUMMY] * num_node\n\n        self.dp = [math.inf] * (num_node + 1)\n        self.dp[0] = -1 * math.inf\n        self.ans = [0] * (num_node + 1) # [0] is dummy\n\n\n    def node_values(self, node_values):\n        self.node_values = [-1] + node_values # [0] is dummy\n\n    def make_edge(self, a, b):\n        self.edges[a].add(b)\n        self.edges[b].add(a)\n\n    def dps(self, node_id, prev=0, depth=0):\n        # process\n        value = self.node_values[node_id]\n        key = bisect.bisect_left(self.dp, value)\n        old_value = self.dp[key]\n        self.dp[key] = value\n        self.ans[node_id] = max(key, self.ans[prev])\n        # recursive\n        for to in self.edges[node_id]:\n            if to == prev:\n                continue\n            self.dps(to, node_id, depth + 1)\n        # recursive back\n        self.dp[key] = old_value\n\ndef __starting_point():\n    num_node = int(input())\n    node_values = list(map(int, input().split()))\n    tree = Tree(num_node, node_values)\n    for i in range(num_node - 1):\n        frm, to = list(map(int, input().split()))\n        tree.make_edge(frm, to)\n    tree.dps(1)\n    for a in tree.ans[1:]:\n        print(a)\n\n\n\n__starting_point()'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 1145445