Christmas Tree

AtCoder

IDarc088_d

Time2000ms

Memory256MB

Difficulty—

Takahashi has decided to make a _Christmas Tree_ for the Christmas party in AtCoder, Inc. A Christmas Tree is a tree with $N$ vertices numbered $1$ through $N$ and $N-1$ edges, whose $i$\-th edge $(1\leq i\leq N-1)$ connects Vertex $a_i$ and $b_i$. He would like to make one as follows: * Specify two non-negative integers $A$ and $B$. * Prepare $A$ _Christmas Paths_ whose lengths are at most $B$. Here, a Christmas Path of length $X$ is a graph with $X+1$ vertices and $X$ edges such that, if we properly number the vertices $1$ through $X+1$, the $i$\-th edge $(1\leq i\leq X)$ will connect Vertex $i$ and $i+1$. * Repeat the following operation until he has one connected tree: * Select two vertices $x$ and $y$ that belong to different connected components. Combine $x$ and $y$ into one vertex. More precisely, for each edge $(p,y)$ incident to the vertex $y$, add the edge $(p,x)$. Then, delete the vertex $y$ and all the edges incident to $y$. * Properly number the vertices in the tree. Takahashi would like to find the lexicographically smallest pair $(A,B)$ such that he can make a Christmas Tree, that is, find the smallest $A$, and find the smallest $B$ under the condition that $A$ is minimized. Solve this problem for him. ## Constraints * $2 \leq N \leq 10^5$ * $1 \leq a_i,b_i \leq N$ * The given graph is a tree. ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $b_1$ : $a_{N-1}$ $b_{N-1}$ [samples]

Samples

Input #1

Output #1

3 2

We can make a Christmas Tree as shown in the figure below:
![image](https://img.atcoder.jp/arc088/96f78221624d6a13628f6052f5db697d.png)

Input #2

Output #2

2 5

Input #3

Output #3

3 4

API Response (JSON)

{
  "problem": {
    "name": "Christmas Tree",
    "description": {
      "content": "Takahashi has decided to make a _Christmas Tree_ for the Christmas party in AtCoder, Inc. A Christmas Tree is a tree with $N$ vertices numbered $1$ through $N$ and $N-1$ edges, whose $i$\\-th edge $(1\\",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc088_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Takahashi has decided to make a _Christmas Tree_ for the Christmas party in AtCoder, Inc.\nA Christmas Tree is a tree with $N$ vertices numbered $1$ through $N$ and $N-1$ edges, whose $i$\\-th edge $(1\\...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments