Farmer John's farm consists of $N$ pastures ($2 \leq N \leq 10^5$) connected by
$N-1$ roads, so that any pasture is reachable from any other pasture. That is,
the farm is a tree. But after 28 years of dealing with the tricky algorithmic
problems that inevitably arise from trees, FJ has decided that a farm in the
shape of a tree is just too complex. He believes that algorithmic problems are
simpler on paths.

Thus, his plan is to partition the set of roads into several paths, and delegate
responsibility for each path to a worthy farm hand. To avoid contention, he
wants each path to be the same length. He wonders for which lengths there exists
such a partition.

More precisely, for each $1 \leq K \leq N-1$, help Farmer John determine whether
the roads can be partitioned into paths of length exactly $K$.

SCORING:
In test cases 2-4 the tree forms a star; at most one vertex has degree
greater than two.Test cases 5-8 satisfy $N\le 10^3$.Test cases 9-15 satisfy no additional constraints..

INPUT FORMAT:
The first line contains a single integer $N$.

The next $N-1$ lines each contain two space-separated integers $a$ and $b$
describing an edge between vertices $a$ and $b$. Each of $a$ and $b$ is in the
range $1 \ldots N$.

OUTPUT FORMAT:
Output a bit string of length $N-1.$ For each $1\le K\le N-1,$ the $K$th bit of
this string from the left should equal one if it is possible to partition the edges of the
tree into paths of length exactly $K$ and $0$ otherwise.

SAMPLE INPUT:
13
1 2
2 3
2 4
4 5
2 6
6 7
6 8
8 9
9 10
8 11
11 12
12 13
SAMPLE OUTPUT:
111000000000
It is possible to partition this tree into paths of length $K$ for $K=1,2,3.$
For $K=3$, a possible set of paths is as follows:

$$13-12-11-8, 10-9-8-6, 7-6-2-3, 5-4-2-1$$

Problem credits: Mark Gordon and Dhruv Rohatgi

见题面

见题面