Problem:
Score : 1500 points

Problem Statement

You are given a tree with N vertices numbered 1, 2, ..., N. The i-th edge connects Vertex a_i and Vertex b_i.
You are also given a string S of length N consisting of 0 and 1. The i-th character of S represents the number of pieces placed on Vertex i.

Snuke will perform the following operation some number of times:

Choose two pieces the distance between which is at least 2, and bring these pieces closer to each other by 1. More formally, choose two vertices u and v, each with one or more pieces, and consider the shortest path between them. Here the path must contain at least two edges. Then, move one piece from u to its adjacent vertex on the path, and move one piece from v to its adjacent vertex on the path.

By repeating this operation, Snuke wants to have all the pieces on the same vertex. Is this possible?
If the answer is yes, also find the minimum number of operations required to achieve it.

Constraints

2 \leq N \leq 2000

|S| = N

S consists of 0 and 1, and contains at least one 1.

1 \leq a_i, b_i \leq N(a_i \neq b_i)

The edges (a_1, b_1), (a_2, b_2), ..., (a_{N - 1}, b_{N - 1}) forms a tree.

Input

Input is given from Standard Input in the following format:

N
S
a_1 b_1
a_2 b_2
:
a_{N - 1} b_{N - 1}

Output

If it is impossible to have all the pieces on the same vertex, print -1. If it is possible, print the minimum number of operations required.

Sample Input 1

7
0010101
1 2
2 3
1 4
4 5
1 6
6 7

Sample Output 1

3

We can gather all the pieces in three operations as follows:

Choose the pieces on Vertex 3 and 5.

Choose the pieces on Vertex 2 and 7.

Choose the pieces on Vertex 4 and 6.

Sample Input 2

7
0010110
1 2
2 3
1 4
4 5
1 6
6 7

Sample Output 2

-1

Sample Input 3

2
01
1 2

Sample Output 3

0

Solution in Python (3.4.3) (judge verdict: Accepted)

import sys
from heapq import nlargest

sys.setrecursionlimit(10000)


def first_dfs(v, p, stones, dists, costs):
    stone = s[v]
    dist = 0
    cost = 0
    children_dists = []
    for u in links[v]:
        if u == p:
            continue
        us, ud, uc = first_dfs(u, v, stones, dists, costs)
        stone += us
        dist += us + ud
        cost += uc
        children_dists.append((us + ud, uc))
    # print(v, p, stone, dist, cost, children_dists)
    if len(children_dists) > 1:
        md, mc = max(children_dists)
        if dist - md < md - mc * 2:
            cost = dist - md + mc
        else:
            cost = dist // 2
        # print(v, p, md, hd, dist, cost)
    stones[v] = stone
    dists[v] = dist
    costs[v] = cost
    return stone, dist, cost


def second_dfs(v, p, pd, pc):
    children_dists = []
    dist = 0

    for u in links[v]:
        if u == p:
            dist += pd
            children_dists.append((pd, pc))
            continue
        ud = stones[u] + dists[u]
        uc = costs[u]
        dist += ud
        children_dists.append((ud, uc))
    # print(v, p, pd, pc, dist, children_dists)

    # only when source and leaf node
    if len(children_dists) == 1:
        u = next(iter(links[v]))
        return second_dfs(u, v, s[v], 0)

    (md1, mc1), (md2, mc2) = nlargest(2, children_dists)
    ret = INF

    hd, m = divmod(dist, 2)
    # print(' ', hd, m, md1, mc1, dist - md1, md1 - mc1 * 2)
    if m == 0 and dist - md1 >= md1 - mc1 * 2:
        ret = min(ret, hd)

    for u in links[v]:
        if u == p:
            continue
        us = stones[u]
        ud = dists[u]
        uc = costs[u]

        # dist=0 node cannot be a center
        if ud == 0:
            continue
        usd = us + ud
        if usd == md1 and uc == mc1:
            md, mc = md2, mc2
        else:
            md, mc = md1, mc1
        td = dist - usd
        if td - md < md - mc * 2:
            vc = td - md + mc
        else:
            vc = td // 2
        result = second_dfs(u, v, td + (all_stones - us), vc)
        ret = min(ret, result)

    return ret


n = int(input())
s = list(map(int, input()))
links = [set() for _ in [0] * n]
for line in sys.stdin:
    a, b = map(int, line.split())
    a -= 1
    b -= 1
    links[a].add(b)
    links[b].add(a)
all_stones = sum(s)
if all_stones == 1:
    print(0)
    exit()
stones = [0] * n  # 1を根とした木の、頂点vの部分木以下（自身含む）にある石の数
dists = [0] * n  # 頂点vの部分木以下にある石 x それぞれのvとの距離 の和
costs = [0] * n  # 頂点v以下の石同士で、最大限頂点vに近づけておくためにできる操作の数
first_dfs(0, -1, stones, dists, costs)
# print(stones)
# print(dists)
# print(costs)
INF = 10 ** 10
ret = second_dfs(0, -1, 0, 0)
print(-1 if ret == INF else ret)