Game on Tree

题意翻译

**题目描述** 给定一棵有根树，结点编号从 $1$ 到 $n$。根结点为 $1$ 号结点。 对于每一次操作，等概率的选择一个**尚未被删去**的结点并将它及其子树全部删去。当所有结点被删除之后，游戏结束；也就是说，删除 $1$ 号结点后游戏即结束。 要求求出删除所有结点的期望操作次数。 **输入格式** 第一行，一个正整数 $n$ 表示结点数量。 接下来 $n-1$ 行每行两个数，表示树上的一条连接 $a_i$ 与 $b_i$ 的边 $(a_i,b_i)$ 保证给定的数据是一棵树。 **输出格式** 输出一个实数，表示期望操作次数。答案误差在 $10^{-6}$ 之内则认为正确。 **样例解释** 在第一个样例中，有两种情况： 一种是直接删除根（即 $1$ 号结点），另一种是先删去 $2$ 号结点，再删除 $1$ 号结点。 操作次数的期望是 $1\times \dfrac12+2\times\dfrac12=1.5$。 在第二个样例中，情况更为复杂。其中有两种情况会将问题转化成第一个样例，而剩下的一种情况会一次全部删除。 操作次数的期望是 $1\times\dfrac13+(1+1.5)\times\dfrac23=\dfrac13+\dfrac53=2$。

题目描述

Momiji has got a rooted tree, consisting of $ n $ nodes. The tree nodes are numbered by integers from $ 1 $ to $ n $ . The root has number $ 1 $ . Momiji decided to play a game on this tree. The game consists of several steps. On each step, Momiji chooses one of the remaining tree nodes (let's denote it by $ v $ ) and removes all the subtree nodes with the root in node $ v $ from the tree. Node $ v $ gets deleted as well. The game finishes when the tree has no nodes left. In other words, the game finishes after the step that chooses the node number $ 1 $ . Each time Momiji chooses a new node uniformly among all the remaining nodes. Your task is to find the expectation of the number of steps in the described game.

输入输出格式

输入格式

The first line contains integer $ n $ $ (1<=n<=10^{5}) $ — the number of nodes in the tree. The next $ n-1 $ lines contain the tree edges. The $ i $ -th line contains integers $ a_{i} $ , $ b_{i} $ $ (1<=a_{i},b_{i}<=n; a_{i}≠b_{i}) $ — the numbers of the nodes that are connected by the $ i $ -th edge. It is guaranteed that the given graph is a tree.

输出格式

Print a single real number — the expectation of the number of steps in the described game. The answer will be considered correct if the absolute or relative error doesn't exceed $ 10^{-6} $ .

输入输出样例

输入样例 #1

2
1 2

输出样例 #1

1.50000000000000000000

输入样例 #2

3
1 2
1 3

输出样例 #2

2.00000000000000000000

说明

In the first sample, there are two cases. One is directly remove the root and another is remove the root after one step. Thus the expected steps are: $ 1×(1/2)+2×(1/2)=1.5 $ In the second sample, things get more complex. There are two cases that reduce to the first sample, and one case cleaned at once. Thus the expected steps are: $ 1×(1/3)+(1+1.5)×(2/3)=(1/3)+(5/3)=2 $