Minimum Diameter Tree
Description
You are given a tree (an undirected connected graph without cycles) and an integer $s$.
Vanya wants to put weights on all edges of the tree so that all weights are non-negative real numbers and their sum is $s$. At the same time, he wants to make the diameter of the tree as small as possible.
Let's define the diameter of a weighed tree as the maximum sum of the weights of the edges lying on the path between two some vertices of the tree. In other words, the diameter of a weighed tree is the length of the longest simple path in the tree, where length of a path is equal to the sum of weights over all edges in the path.
Find the minimum possible diameter that Vanya can get.
The first line contains two integer numbers $n$ and $s$ ($2 \leq n \leq 10^5$, $1 \leq s \leq 10^9$) — the number of vertices in the tree and the sum of edge weights.
Each of the following $n−1$ lines contains two space-separated integer numbers $a_i$ and $b_i$ ($1 \leq a_i, b_i \leq n$, $a_i \neq b_i$) — the indexes of vertices connected by an edge. The edges are undirected.
It is guaranteed that the given edges form a tree.
Print the minimum diameter of the tree that Vanya can get by placing some non-negative real weights on its edges with the sum equal to $s$.
Your answer will be considered correct if its absolute or relative error does not exceed $10^{-6}$.
Formally, let your answer be $a$, and the jury's answer be $b$. Your answer is considered correct if $\frac {|a-b|} {max(1, b)} \leq 10^{-6}$.
Input
The first line contains two integer numbers $n$ and $s$ ($2 \leq n \leq 10^5$, $1 \leq s \leq 10^9$) — the number of vertices in the tree and the sum of edge weights.
Each of the following $n−1$ lines contains two space-separated integer numbers $a_i$ and $b_i$ ($1 \leq a_i, b_i \leq n$, $a_i \neq b_i$) — the indexes of vertices connected by an edge. The edges are undirected.
It is guaranteed that the given edges form a tree.
Output
Print the minimum diameter of the tree that Vanya can get by placing some non-negative real weights on its edges with the sum equal to $s$.
Your answer will be considered correct if its absolute or relative error does not exceed $10^{-6}$.
Formally, let your answer be $a$, and the jury's answer be $b$. Your answer is considered correct if $\frac {|a-b|} {max(1, b)} \leq 10^{-6}$.
Samples
4 3
1 2
1 3
1 4
2.000000000000000000
6 1
2 1
2 3
2 5
5 4
5 6
0.500000000000000000
5 5
1 2
2 3
3 4
3 5
3.333333333333333333
Note
In the first example it is necessary to put weights like this:
It is easy to see that the diameter of this tree is $2$. It can be proved that it is the minimum possible diameter.
In the second example it is necessary to put weights like this:
