A10823. The Winds of Winter

Given a rooted tree with $n$ nodes. The Night King removes exactly one node from the tree and all the edges associated with it. Doing this splits the tree and forms a forest. The node which is removed is not a part of the forest.

The root of a tree in the forest is the node in that tree which does not have a parent. We define the strength of the forest as the size of largest tree in forest.

Jon Snow wants to minimize the strength of the forest. To do this he can perform the following operation at most once.

He removes the edge between a node and its parent and inserts a new edge between this node and any other node in forest such that the total number of trees in forest remain same.

For each node $v$ you need to find the minimum value of strength of the forest formed when node $v$ is removed.

The first line of the input contains an integer $n$ ( $1<=n<=10^{5}$ ) — the number of vertices in the tree. Each of the next $n$ lines contains a pair of vertex indices $u_{i}$ and $v_{i}$ ( $1<=u_{i},v_{i}<=n$ ) where $u_{i}$ is the parent of $v_{i}$ . If $u_{i}=0$ then $v_{i}$ is the root.

Print $n$ line each containing a single integer. The $i$ -th of them should be equal to minimum value of strength of forest formed when $i$ -th node is removed and Jon Snow performs the operation described above at most once.

The tree for first test case is depicted below.  When you remove the first node, the tree splits to form the following forest. The strength of this forest is $4$ .  Jon Snow now changes the parent of vertex $10$ from $5$ to $3$ . The strength of forest now becomes $3$ .