You are given a tree$$$^{\text{∗}}$$$ with $$$n$$$ vertices. Over time, each vertex $$$i$$$ ($$$1 \le i \le n$$$) has a probability of $$$\frac{p_i}{q_i}$$$ of falling. Determine the expected value of the number of unordered pairs$$$^{\text{†}}$$$ of distinct vertices that become leaves$$$^{\text{‡}}$$$ in the resulting forest$$$^{\text{§}}$$$, modulo $$$998\,244\,353$$$.
Note that when vertex $$$v$$$ falls, it is removed along with all edges connected to it. However, adjacent vertices remain unaffected by the fall of $$$v$$$.
$$$^{\text{∗}}$$$A tree is a connected graph without cycles.
$$$^{\text{†}}$$$An unordered pair is a collection of two elements where the order in which the elements appear does not matter. For example, the unordered pair $$$(1, 2)$$$ is considered the same as $$$(2, 1)$$$.
$$$^{\text{‡}}$$$A leaf is a vertex that is connected to exactly one edge.
$$$^{\text{§}}$$$A forest is a graph without cycles
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$).
The $$$i$$$-th line of the following $$$n$$$ lines contains two integers $$$p_i$$$ and $$$q_i$$$ ($$$1 \le p_i < q_i < 998\,244\,353$$$).
Each of the following $$$n - 1$$$ lines contains two integers $$$u$$$ and $$$v$$$ ($$$1 \le u, v \le n$$$, $$$u \neq v$$$) — the indices of the vertices connected by an edge.
It is guaranteed that the given edges form a tree.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.
For each test case, output a single integer — the expected value of the number of unordered pairs of distinct vertices that become leaves in the resulting forest modulo $$$998\,244\,353$$$.
Formally, let $$$M = 998\,244\,353$$$. It can be shown that the exact answer can be expressed as an irreducible fraction $$$\frac{p}{q}$$$, where $$$p$$$ and $$$q$$$ are integers and $$$q \not \equiv 0 \pmod{M}$$$. Output the integer equal to $$$p \cdot q^{-1} \bmod M$$$. In other words, output such an integer $$$x$$$ that $$$0 \le x < M$$$ and $$$x \cdot q \equiv p \pmod{M}$$$.
511 231 21 21 21 22 331 31 51 31 22 31998244351 998244352610 177 136 112 1010 195 134 33 61 43 53 2
0 623902721 244015287 0 799215919
110282508078 551568452894311255 989959022893400641 913415297460925436 80190898594460427 171411253997964895 998217862770266391 885105593591419316 976424827606447024 863339056940224886 9942445539 59 69 88 73 61 57 48 104 2
486341067
In the first test case, only one vertex is in the tree, which is not a leaf, so the answer is $$$0$$$.
In the second test case, the tree is shown below.
Vertices that have not fallen are denoted in bold. Let us examine the following three cases:
All remaining cases contain no unordered pairs of distinct leaf vertices. Hence, the answer is $$$\frac{1 + 1 + 1}{8} = \frac{3}{8}$$$, which is equal to $$$623\,902\,721$$$ modulo $$$998\,244\,353$$$.