D. You Are Given a Tree
data structures; dp; trees
A tree is an undirected graph with exactly one simple path between each pair of vertices. We call a set of simple paths \(k\)-valid if each vertex of the tree belongs to no more than one of these paths (including endpoints) and each path consists of exactly \(k\) vertices.You are given a tree with \(n\) vertices. For each \(k\) from \(1\) to \(n\) inclusive find what is the maximum possible size of a \(k\)-valid set of simple paths.
The first line of the input contains a single integer \(n\) (\(2 \le n \le 100\,000\)) — the number of vertices in the tree.Then following \(n - 1\) lines describe the tree, each of them contains two integers \(v\), \(u\) (\(1 \le v, u \le n\)) — endpoints of the corresponding edge.It is guaranteed, that the given graph is a tree.
Output \(n\) numbers, the \(i\)-th of which is the maximum possible number of paths in an \(i\)-valid set of paths.
One way to achieve the optimal number of paths for the second sample is illustrated in the following picture: