Logical Operations on Tree

AtCoder

IDarc121_f

Time2000ms

Memory256MB

Difficulty—

Given is a tree with $N$ vertices numbered $1$ through $N$. The $i$\-th edge connects Vertices $a_i$ and $b_i$. Snuke will label each vertex with `0` or `1` and each edge with `AND` or `OR`. Among the $2^{2N-1}$ ways to label the vertices and edges, find the number of ways that satisfy the following condition, modulo $998244353$. Condition: There exists a sequence of $N-1$ operations ending up with one vertex labeled `1`, where each operation consists of the steps below. * Choose one edge and contract it. Here, let $x$ and $y$ be the labels of the erased vertices and $\mathrm{op}$ be the label of the erased edge. * If $\mathrm{op}$ is `AND`, label the new vertex with $\mathrm{AND}(x,y)$; if $\mathrm{op}$ is `OR`, label the new vertex with $\mathrm{OR}(x,y)$. ## Constraints * All values in input are integers. * $2 \leq N \leq 10^{5}$ * $1 \leq a_i, b_i \leq N$ * The given graph is a tree. ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $b_1$ $\vdots$ $a_{N-1}$ $b_{N-1}$ [samples] ## Notes * The operation $\mathrm{AND}$ is defined as follows: $\mathrm{AND}(0,0)=(0,1)=(1,0)=0,\mathrm{AND}(1,1)=1$. * The operation $\mathrm{OR}$ is defined as follows: $\mathrm{OR}(1,1)=(0,1)=(1,0)=1,\mathrm{OR}(0,0)=0$. * When contracting the edge connecting Vertex $s$ and Vertex $t$, we merge the two vertices while removing that edge. After the contraction, there is an edge connecting the new vertex and vertex $u$ if and only if there was an edge connecting $s$ and $u$ or connecting $t$ and $u$ before the contraction.

Samples

Input #1

2
1 2

Output #1

Input #2

Output #2

283374562

*   Remember to print the count modulo $998244353$.

API Response (JSON)

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    "name": "Logical Operations on Tree",
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      "content": "Given is a tree with $N$ vertices numbered $1$ through $N$. The $i$\\-th edge connects Vertices $a_i$ and $b_i$. Snuke will label each vertex with `0` or `1` and each edge with `AND` or `OR`. Among the",
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      "statement_type": "Markdown",
      "content": "Given is a tree with $N$ vertices numbered $1$ through $N$.\nThe $i$\\-th edge connects Vertices $a_i$ and $b_i$.\nSnuke will label each vertex with `0` or `1` and each edge with `AND` or `OR`. Among the...",
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