Path-Digging Animals (PDA) is a new cooperative game for two players that is gaining popularity among programmers. The game is played on a board having $$$N$$$ holes and $$$N-1$$$ bidirectional tunnels. The tunnels form an undirected tree, which means that there is a unique simple path between each pair of holes.
A route in PDA is a simple path between two different holes. Since tunnels are bidirectional, the direction of a route is not relevant: the (unique) route from hole $$$U$$$ to hole $$$V$$$ and the (unique) route from hole $$$V$$$ to hole $$$U$$$ are the same route.
Players use animal-shaped pieces for playing PDA@. In each round of the game, each player independently chooses a route which is traversed by their animal, and both players score as many points as the number of tunnels that their route s have in common.
Right now Alicia and Bruno are at a board game party in Chile playing PDA, and they want to analyze their scoring possibilities. They would like to know, for each integer $$$k$$$ from $$$1$$$ to $$$N-1$$$, how many ordered pairs of route s $$$(A,B)$$$ there are such that Alicia and Bruno score exactly $$$k$$$ points if Alicia chooses route $$$A$$$ and Bruno chooses route $$$B$$$.
The first line contains an integer $$$N$$$ $$$(2 \leq N \leq 2 \times 10^{5})$$$ indicating the number of holes. Each hole is identified by a distinct integer from $$$1$$$ to $$$N$$$.
Each of the next $$$N-1$$$ lines contains two integers $$$U$$$ and $$$V$$$ ($$$1 \leq U, V \leq N$$$ and $$$U \neq V$$$), representing that there is a bidirectional tunnel between holes $$$U$$$ and $$$V$$$. It is guaranteed that the tunnels form an undirected tree.
Output a single line with $$$N-1$$$ integers $$${P_1}, {P_2}, \ldots, {P_{N-1}}$$$, where $$$P_k$$$ indicates the number of ordered pairs of route s $$$(A,B)$$$ such that Alicia and Bruno score exactly $$$k$$$ points if Alicia chooses route $$$A$$$ and Bruno chooses route $$$B$$$. Because these numbers can be very large, output the remainder of dividing each of them by $$$998244353$$$.
31 32 3
6 1
Explanation for example 1
There are three possibilities for Alicia's route $$$A$$$: the route between holes $$$1$$$ and $$$3$$$ (formed by the single tunnel between these holes), the route between holes $$$2$$$ and $$$3$$$ (again a single tunnel), and the route between holes $$$2$$$ and $$$1$$$ (containing both tunnels). The same three possibilities are available for Bruno's route $$$B$$$. The table below shows the score for each combination. As can be seen, $$$6$$$ ordered pairs of route s yield a score of $$$1$$$ point, while $$$1$$$ pair gives $$$2$$$ points. Applying the modulo operation does not change these results.
| Bruno | ||||
| 1–3 | 2–3 | 2–3–1 | ||
| 1–3 | 1 | 0 | 1 | |
| Alicia | 2–3 | 0 | 1 | 1 |
| 2–3–1 | 1 | 1 | 2 | |
