Tree Operation

You are given a tree with $N$ vertices. Initially, the number written on every vertex is $0$.

For each vertex, you may perform the following operation at most once.

Choose one vertex and change the number written on it to the minimum number written on an adjacent vertex, plus $1$.

If the chosen vertex has no adjacent vertices, the minimum number among adjacent vertices is considered to be $0$.

Find the maximum possible sum of all numbers written on the tree after performing operations in a suitable order.

The input is given in the following format.

$N$ $u_1\ v_1$ $u_2\ v_2$ $\cdots$ $u_{N-1}\ v_{N-1}$

The given graph is a tree with $N$ vertices. The vertices are numbered from $1$ to $N$.

Print the maximum possible sum of all numbers written on the tree.

• $1 \le N \le 200\,000$.
• $1 \le u_i, v_i \le N$.
• $u_i \ne v_i$.
• The given graph is a tree.