F - Distance Sums 2

Score : $500$ points

Problem Statement

Given is a tree with $N$ vertices. The vertices are numbered $1,2,\ldots ,N$, and the $i$-th edge is an undirected edge connecting Vertices $u_i$ and $v_i$.

For each integer $i\,(1 \leq i \leq N)$, find $\sum_{j=1}^{N}dis(i,j)$.

Here, $dis(i,j)$ denotes the minimum number of edges that must be traversed to go from Vertex $i$ to Vertex $j$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq u_i < v_i \leq N$
• The given graph is a tree.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_{N-1}$ $v_{N-1}$

Output

Print $N$ lines.

The $i$-th line should contain $\sum_{j=1}^{N}dis(i,j)$.

3
1 2
2 3

3
2
3

We have:

$dis(1,1)+dis(1,2)+dis(1,3)=0+1+2=3$,

$dis(2,1)+dis(2,2)+dis(2,3)=1+0+1=2$,

$dis(3,1)+dis(3,2)+dis(3,3)=2+1+0=3$.

2
1 2

1
1

6
1 6
1 5
1 3
1 4
1 2

5
9
9
9
9
9