A15295 | Imitating the Key Tree

Pak Chanek has a tree called the key tree. This tree consists of $N$ vertices and $N-1$ edges. The edges of the tree are numbered from $1$ to $N-1$ with edge $i$ connecting vertices $U_i$ and $V_i$ . Initially, each edge of the key tree does not have a weight.

Formally, a path with length $k$ in a graph is a sequence $[v_1, e_1, v_2, e_2, v_3, e_3, \ldots, v_k, e_k, v_{k+1}]$ such that:

- For each $i$ , $v_i$ is a vertex and $e_i$ is an edge.
- For each $i$ , $e_i$ connects vertices $v_i$ and $v_{i+1}$ .

A circuit is a path that starts and ends on the same vertex.

A path in a graph is said to be simple if and only if the path does not use the same edge more than once. Note that a simple path can use the same vertex more than once.

The cost of a simple path in a weighted graph is defined as the maximum weight of all edges it traverses.

Count the number of distinct undirected weighted graphs that satisfy the following conditions:

- The graph has $N$ vertices and $2N-2$ edges.
- For each pair of different vertices $(x, y)$ , there exists a simple circuit that goes through vertices $x$ and $y$ in the graph.
- The weight of each edge in the graph is an integer between $1$ and $2N-2$ inclusive. Each edge has distinct weights.
- The graph is formed in a way such that there is a way to assign a weight $W_i$ to each edge $i$ in the key tree that satisfies the following conditions:
- For each pair of edges $(i, j)$ , if $i<j$ , then $W_i<W_j$ .
- For each pair of different vertex indices $(x, y)$ , the cost of the only simple path from vertex $x$ to $y$ in the key tree is equal to the minimum cost of a simple circuit that goes through vertices $x$ and $y$ in the graph.
- Note that the graph is allowed to have multi-edges, but is not allowed to have self-loops.

Print the answer modulo $998\,244\,353$ .

Two graphs are considered distinct if and only if there exists a triple $(a, b, c)$ such that there exists an edge that connects vertices $a$ and $b$ with weight $c$ in one graph, but not in the other.

The first line contains a single integer $N$ ( $2 \le N \le 10^5$ ) — the number of vertices in the key tree.

The $i$ -th of the next $N-1$ lines contains two integers $U_i$ and $V_i$ ( $1 \le U_i, V_i \le N$ ) — an edge connecting vertices $U_i$ and $V_i$ . The graph in the input is a tree.

An integer representing the number of distinct undirected weighted graphs that satisfy the conditions of the problem modulo $998\,244\,353$ .