Portable Gate

AtCoder

IDarc179_d

Time2000ms

Memory256MB

Difficulty—

You are given a tree with $N$ vertices numbered $1, 2, \dots, N$. The $i$\-th edge connects vertices $u_i$ and $v_i$ bidirectionally. Initially, all vertices are painted white. To efficiently visit all vertices of this tree, Alice has invented a magical gate. She uses one piece and one gate to travel according to the following procedure. First, she chooses a vertex and places both the piece and the gate on that vertex. Then, she repeatedly performs the following operations until all vertices are painted black. * Choose one of the following actions: 1. Paint the vertex where the piece is placed black. 2. Choose a vertex adjacent to the vertex where the piece is placed and move the piece to that vertex. The cost of this action is $1$. 3. Move the piece to the vertex where the gate is placed. 4. Move the gate to the vertex where the piece is placed. Note that only the second action incurs a cost. It can be proved that it is possible to paint all vertices black in a finite number of operations. Find the minimum total cost required. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq u_i, v_i \leq N$ * The given graph is a tree. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $u_1$ $v_1$ $\vdots$ $u_{N-1}$ $v_{N-1}$ [samples]

Samples

Input #1

Output #1

3

Here is an example of Alice's procedure. Let $(u, v)$ denote the state where the piece is at vertex $u$ and the gate is at vertex $v$.

*   Place the piece and the gate at vertex $4$.
    *   The state is now $(4, 4)$.
*   Perform action $1$.
    *   Vertex $4$ is painted black.
    *   The state is now $(4, 4)$.
*   Perform action $2$ and move the piece to vertex $1$.
    *   This costs $1$.
    *   The state is now $(1, 4)$.
*   Perform action $1$.
    *   Vertex $1$ is painted black.
*   Perform action $4$.
    *   The state is now $(1, 1)$.
*   Perform action $2$ and move the piece to vertex $2$.
    *   This costs $1$.
    *   The state is now $(2, 1)$.
*   Perform action $1$.
    *   Vertex $2$ is painted black.
*   Perform action $3$.
    *   The state is now $(1, 1)$.
*   Perform action $2$ and move the piece to vertex $3$.
    *   This costs $1$.
    *   The state is now $(3, 1)$.
*   Perform action $1$.
    *   Vertex $3$ is painted black.
    *   All vertices are now painted black, so the procedure ends.

The total cost of performing action $2$ is $3$, and there is no procedure with a smaller cost.

Input #2

Output #2

API Response (JSON)

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      "statement_type": "Markdown",
      "content": "You are given a tree with $N$ vertices numbered $1, 2, \\dots, N$. The $i$\\-th edge connects vertices $u_i$ and $v_i$ bidirectionally.\nInitially, all vertices are painted white.\nTo efficiently visit al...",
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