Tree Coloring

- 把一棵 $n$ 个点的有根树涂 $n$ 种颜色，使得任意节点颜色不等于其父亲的颜色减一，且每个点涂的颜色互不相同。 - 求方案数取模 $998244353$

You are given a rooted tree consisting of $ n $ vertices numbered from $ 1 $ to $ n $ . The root of the tree is the vertex $ 1 $ . You have to color all vertices of the tree into $ n $ colors (also numbered from $ 1 $ to $ n $ ) so that there is exactly one vertex for each color. Let $ c_i $ be the color of vertex $ i $ , and $ p_i $ be the parent of vertex $ i $ in the rooted tree. The coloring is considered beautiful if there is no vertex $ k $ ( $ k > 1 $ ) such that $ c_k = c_{p_k} - 1 $ , i. e. no vertex such that its color is less than the color of its parent by exactly $ 1 $ . Calculate the number of beautiful colorings, and print it modulo $ 998244353 $ .

The first line contains one integer $ n $ ( $ 2 \le n \le 250000 $ ) — the number of vertices in the tree. Then $ n-1 $ lines follow, the $ i $ -th line contains two integers $ x_i $ and $ y_i $ ( $ 1 \le x_i, y_i \le n $ ; $ x_i \ne y_i $ ) denoting an edge between the vertex $ x_i $ and the vertex $ y_i $ . These edges form a tree.

Print one integer — the number of beautiful colorings, taken modulo $ 998244353 $ .

5
1 2
3 2
4 2
2 5

42

5
1 2
2 3
3 4
4 5

53

20
20 19
20 4
12 4
5 8
1 2
20 7
3 10
7 18
11 8
9 10
17 10
1 15
11 16
14 11
18 10
10 1
14 2
13 17
20 6

955085064