Mashtali vs AtCoder solution codeforces

Mashtali vs AtCoder solution codeforces

After many unsuccessful tries, Mashtali decided to copy modify an AtCoder problem. So here is his copied new problem:

There is a tree with nn vertices and some non-empty set of the vertices are pinned to the ground.

Two players play a game against each other on the tree. They alternately perform the following action:

  • Remove an edge from the tree, then remove every connected component that has no pinned vertex.The player who cannot move loses(every edge has been deleted already). Mashtali vs AtCoder solution codeforces

You are given the tree, but not the set of the pinned vertices. Your task is to determine, for each kk, the winner of the game, if only the vertices 1,2,3,,k1,2,3,…,k are pinned and both players play optimally.

Input Mashtali vs AtCoder solution codeforces

The first line of input contains an integer nn — the number of vertices (1n31051≤n≤3⋅105).

The ii-th of the following n1n−1 lines contains two integers ui,viui,vi (1ui,vin1≤ui,vi≤nuiviui≠vi) — the endpoints of the ii-th edge. It’s guaranteed that these edges form a tree.

Output

Print a string of length nn. The ii-th character should be ‘1’ if the first player wins the ii-th scenario, and ‘2’ otherwise.

Examples

input Mashtali vs AtCoder solution codeforces

Copy
5
1 2
2 3
2 4
4 5

output

Copy
11122

input Mashtali vs AtCoder solution codeforces

Copy
5
1 2
2 3
1 4
4 5

output

Copy
21122

input Mashtali vs AtCoder solution codeforces

Copy
6
1 2
2 4
5 1
6 3
3 2

output

Copy
111111

input Mashtali vs AtCoder solution codeforces

Copy
7
1 2
3 7
4 6
2 3
2 4
1 5

output Mashtali vs AtCoder solution codeforces

Copy
2212222
Note

Below you can see the tree in the first sample :

If k=1k=1 then the first player can cut the edge (1,2)(1,2).

If k=2k=2 or k=3k=3, the first player can cut the edge (2,4)(2,4), after that only the edges (1,2)(1,2) and (2,3)(2,3) remain. After the second players move, there will be a single edge left for the first player to cut. So first player wins.

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