P4748 [CERC2017] Justified Jungle

As you probably know, a tree is a graph consisting of $n$ nodes and $n - 1$ undirected edges in which any two nodes are connected by exactly one path. A forest is a graph consisting of one or more trees. In other words, a graph is a forest if every connected component is a tree. A forest is justified if all connected components have the same number of nodes. Given a tree $G$ consisting of n nodes, find all positive integers $k$ such that a justified forest can be obtained by erasing exactly $k$ edges from $G$. Note that erasing an edge never erases any nodes. In particular when we erase all $n - 1$ edges from $G$, we obtain a justified forest consisting of $n$ one-node components.

The first line contains an integer $n(2 \le n \le 1 000 000)$ — the number of nodes in $G$. The $k-th$ of the following $n - 1$ lines contains two different integers $a_k$ and $b_k(1 \le a_k, b_k \le n)$ — the endpoints of the $k-th$ edge.

The first line should contain all wanted integers $k$, in increasing order.