F - Components

Score : $500$ points

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.

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 $x=i$.

4
1 2
2 3
3 4

10
5
0
0

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

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

2
1 2

3
0

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

140
281
352
195
52
3
0
0
0
0