Be Careful

You are given a rooted tree with $n$ vertices, where the root is vertex $1$. A vertex is a leaf if it is not the root vertex and its degree is exactly $1$.

The figure corresponds to the sample tests, where the leaves are marked red.

Let $\operatorname{mex}(S)$ be the minimal non-negative integer that is not present in $S$. For example, $\operatorname{mex}\{0, 1, 3, 4\} = 2$, $\operatorname{mex}\{2, 3\} = 0$, $\operatorname{mex}\varnothing = 0$.

Let $m$ be the number of leaves in the given tree. You will perform the following procedure:

1. For every leaf vertex $u$, write any integer from $\{0, 1, 2, \ldots, n\}$ to the vertex $u$.
2. For every non-leaf vertex $u$, the integer written in $u$ will be the $\operatorname{mex}$ of the integers written in all the sons of vertex $u$.

For example, for the first tree in the figure, if you write $0$ to vertex $4$ and $3$ to vertex $5$, then vertex $2$ gets $\operatorname{mex}\{0\} = 1$, vertex $3$ gets $\operatorname{mex}\{3\} = 0$, and vertex $1$ gets $\operatorname{mex}\{1, 0\} = 2$.

In total, there are $(n+1)^m$ ways to fill the tree. For every $k \in \{0, 1, 2, \ldots, n\}$, count the ways to fill the tree so that the number written in vertex $1$ is exactly $k$. Since the numbers can be huge, output them modulo $998\,244\,353$.