Trimming the Tree

A tree with $N$ vertices and $N-1$ edges is given. Each vertex of the tree is labeled with a distinct number from $1$ to $N$. The length of every edge in the tree is 1.

The diameter of a tree is the maximum distance among all pairs of distinct vertices in the tree.

Donggeon can perform the following operation on the tree. One operation consists of the three steps below, performed in order.

1. Choose two adjacent vertices $s$ and $t$ in the tree and delete the edge connecting $s$ and $t$. After the deletion, let $S$ be the tree containing $s$ and $T$ be the tree containing $t$.
2. Choose a vertex $p$ in tree $S$.
3. Add an edge connecting $p$ and $t$ to join the two trees back into one tree.

When Donggeon can perform the operation at most $K$ times, find the diameter of the longest-diameter tree that Donggeon can obtain.

The first line contains two integers $N$ and $K$, separated by a space. ($2 \le N \le 100\,000$, $0 \le K \le 100\,000$)

From the second line onward, the next $N-1$ lines each contain two integers $u$ and $v$, separated by a space. This means there is an edge between vertex $u$ and vertex $v$. ($1 \le u, v \le N$, $u \ne v$)

Every edge of the tree is given exactly once.

On the first line, print the diameter of the longest-diameter tree that Donggeon can obtain.

Perform one operation on the tree below.

1. Choose two adjacent vertices $2$ and $4$ and delete the edge connecting $2$ and $4$. After the deletion, let $S$ be the tree containing $2$ and $T$ be the tree containing $4$.

2. Choose vertex $3$ in tree $S$.
3. Add an edge connecting $3$ and $4$ to join the two trees back into one tree.