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Ex - Balanced Tree

Score : $600$ points

Problem Statement

You are given a tree $T$ with $N$ vertices. The $i$-th edge connects vertices $A_i$ and $B_i$.

Construct an $N$-vertex rooted tree $R$ that satisfies the following conditions.

• For all integer pairs $(x,y)$ such that $1 \leq x < y \leq N$,
  • if the lowest common ancestor in $R$ of vertices $x$ and $y$ is vertex $z$, then vertex $z$ in $T$ is on the simple path between vertices $x$ and $y$.
• In $R$, for all vertices $v$ except for the root, the number of vertices of the subtree rooted at $v$, multiplied by $2$, does not exceed the number of vertices of the subtree rooted at the parent of $v$.

We can prove that such a rooted tree always exists.

Constraints

• $1 \leq N \leq 10^5$
• $1\leq A_i,B_i \leq N$
• All values in the input are integers.
• 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

Let $R$ be a rooted tree that satisfies the conditions in the Problem Statement. Suppose that the parent of vertex $i$ in $R$ is vertex $p_i$. (If $i$ is the root, let $p_i=-1$.)
Print $N$ integers $p_1,\ldots,p_N$, with spaces in between.

4
1 2
2 3
3 4

2 -1 4 2

For example, the lowest common ancestor in $R$ of vertices $1$ and $3$ is vertex $2$; in $T$, vertex $2$ is on the simple path between vertices $1$ and $3$.
Also, for example, in $R$, the subtree rooted at vertex $4$ has two vertices, and the number multiplied by two does not exceed the number of vertices of the subtree rooted at vertex $2$, which has $4$ vertices.

5
1 2
1 3
1 4
1 5

-1 1 1 1 1