Problem Statement

You are given a tree with $N$ vertices. The vertices are numbered $1$ through $N$, and the $i$-th edge connects vertex $a_i$ and vertex $b_i$.

Solve the following problem for each $x=1,2,\\ldots,N$:

• There are $(2^N-1)$ non-empty subsets $V$ of the tree's vertices. Find the number, modulo $998244353$, of such $V$ that the subgraph induced by $V$ has exactly $x$ connected components.

What is an induced subgraph? Let $S$ be the subset of the vertices of a graph $G$; then the subgraph of $G$ induced by $S$ is a graph whose vertex set is $S$ and whose edge set consists of all edges of $G$ whose both ends are contained in $S$.

Constraints

• $1 \\leq N \\leq 5000$
• $1 \\leq a_i \\lt b_i \\leq N$
• The given graph is a tree.

Input

The input is given from Standard Input in the following format:

$N$ $a_1$ $b_1$ $\\vdots$ $a_{N-1}$ $b_{N-1}$

Output

Print $N$ lines.
The $i$-th line should contain the answer for $x=i$.

Sample Input 1

4
1 2
2 3
3 4

Sample Output 1

10
5
0
0

The induced subgraph will have two connected components in the following five cases and one in the others.

• $V = \\{1,2,4\\}$
• $V = \\{1,3\\}$
• $V = \\{1,3,4\\}$
• $V = \\{1,4\\}$
• $V = \\{2,4\\}$

Sample Input 2

2
1 2

Sample Output 2

3
0

Sample Input 3

10
3 4
3 6
6 9
1 3
2 4
5 6
6 10
1 8
5 7

Sample Output 3

140
281
352
195
52
3
0
0
0
0