Meetings 2

There are N islands where beavers live. The islands are numbered from 1 through N. These islands are connected by N − 1 bidirectional bridges. The bridges are numbered from 1 through N − 1. The bridge i connects the island Ai and the island Bi. It is possible to travel between any islands using some bridges. Each island is inhabited by a beaver.

Sometimes, beavers living in some islands gather in an island, and have a meeting. When the attendees of a meeting are fixed, one of the islands satisfying the following condition is chosen as the location of the meeting:

The sum of the numbers of bridges the attendees have to pass through to gather in the chosen island is minimized.

Here, when the attendees of a meeting are fixed, each attendee of the meeting moves from his/her island of residence to the location of the meeting by passing through the minimum number of bridges.

The attendees of a meeting are looking forward to it very much if there are many candidates for the location of the meeting. When the attendees of a meeting are fixed, the expectation score of the meeting is calculated as the number of candidate islands for the location of the meeting satisfying the above condition. For each integer j between 1 and N, inclusive, you want to know the maximum expectation score of a meeting where j beavers will attend.

Write a program which, given the information of the islands, calculates, for each number of attendees, the maximum expectation score of a meeting.

Read the following data from the standard input. Given values are all integers.

N
A1 B1
.
.
.
AN−1 BN−1

Write N lines to the standard output. In the j-th line (1 ≤ j ≤ N), output the maximum expectation score of a meeting with j attendees.

• 1 ≤ N ≤ 200 000.
• 1 ≤ Ai ≤ N (1 ≤ i ≤ N − 1).
• 1 ≤ Bi ≤ N (1 ≤ i ≤ N − 1).
• Ai ≠ Bi (1 ≤ i ≤ N − 1).
• It is possible to travel between any islands using some bridges.