Tree Inversion

AtCoder

IDabc337_g

Time2000ms

Memory256MB

Difficulty—

You are given a tree $T$ with $N$ vertices: vertex $1$, vertex $2$, $\ldots$, vertex $N$. The $i$\-th edge $(1\leq i\lt N)$ connects vertices $u_i$ and $v_i$. For a vertex $u$ in $T$, define $f(u)$ as follows: * $f(u)\coloneqq$ The number of pairs of vertices $v$ and $w$ in $T$ that satisfy the following two conditions: * Vertex $w$ is contained in the path connecting vertices $u$ and $v$. * $v\lt w$. Here, vertex $w$ is contained in the path connecting vertices $u$ and $v$ also when $u=w$ or $v=w$. Find the values of $f(1),f(2),\ldots,f(N)$. ## Constraints * $2\leq N\leq2\times10^5$ * $1\leq u_i\leq N\ (1\leq i\leq N)$ * $1\leq v_i\leq N\ (1\leq i\leq N)$ * The given graph is a tree. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_{N-1}$ $v_{N-1}$ [samples]

Samples

Input #1

Output #1

0 1 3 4 8 9 15

The given tree is as follows:
![image](https://img.atcoder.jp/abc337/f02c6287abee93272dda64faf9499a81.png)
For example, $f(4)=4$. Indeed, for $u=4$, four pairs $(v,w)=(1,2),(1,4),(2,4),(3,4)$ satisfy the conditions.

Input #2

Output #2

36 29 32 29 48 37 45 37 44 42 33 36 35 57 35

The given tree is as follows:
![image](https://img.atcoder.jp/abc337/e59ed095480b6f8ec4ecfc212f14e635.png)
The value of $f(14)$ is $57$, which is the inversion number of the sequence $(14,9,1,6,12,2,15,4,11,13,3,8,10,7,5)$.

Input #3

Output #3

20 20 41 20 21 20 28 28 43 44 36 63 40 46 34 40 59 28 53 53 66 42 62 63

API Response (JSON)

{
  "problem": {
    "name": "Tree Inversion",
    "description": {
      "content": "You are given a tree $T$ with $N$ vertices: vertex $1$, vertex $2$, $\\ldots$, vertex $N$. The $i$\\-th edge $(1\\leq i\\lt N)$ connects vertices $u_i$ and $v_i$. For a vertex $u$ in $T$, define $f(u)$ as",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc337_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a tree $T$ with $N$ vertices: vertex $1$, vertex $2$, $\\ldots$, vertex $N$. The $i$\\-th edge $(1\\leq i\\lt N)$ connects vertices $u_i$ and $v_i$.\nFor a vertex $u$ in $T$, define $f(u)$ as...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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