Cascade Centrality

Given an undirected graph $G=(V,E)$, the cascade centrality of node $i$ in $V$ is defined as $1 + \sum_{j \in V \setminus \{i\}} \sum_{P \in P_{ij}} \frac{1}{\chi_P},$ where $P_{ij}$ is the set of all simple paths from node $i$ to node $j$. The degree sequence product $\chi_P$ of a path is the product of the degrees of all nodes along the path, including the ending node but excluding the starting node.

Since $G$ is a tree, $P_{ij}$ always contains exactly one path. Find the mean of the cascade centralities of the nodes in $G$.

The first line contains an integer $N$ $(1 \leq N \leq 100)$, the number of nodes in the tree. Each of the next $N-1$ lines contains two space-separated integers $u_i$ and $v_i$ $(1 \leq u_i, v_i \leq N)$, denoting an undirected edge between nodes $u_i$ and $v_i$. No edge connects a node to itself, and there is at most one edge between any pair of nodes. The given graph is connected and has no cycles.

Print the mean of the cascade centralities of the nodes in the graph. Your answer is accepted if it differs from the judge answer by at most $10^{-6}$ in relative or absolute error.