CF1394D.Boboniu and Jianghu

Since Boboniu finished building his Jianghu, he has been doing Kungfu on these mountains every day.

Boboniu designs a map for his $n$ mountains. He uses $n-1$ roads to connect all $n$ mountains. Every pair of mountains is connected via roads.

For the $i$-th mountain, Boboniu estimated the tiredness of doing Kungfu on the top of it as $t_i$. He also estimated the height of each mountain as $h_i$.

A path is a sequence of mountains $M$ such that for each $i$ ($1 \le i \lt |M|$), there exists a road between $M_i$ and $M_{i+1}$. Boboniu would regard the path as a challenge if for each $i$ ($1\le i \lt |M|$), $h_{M_i}\le h_{M_{i+1}}$.

Boboniu wants to divide all $n-1$ roads into several challenges. Note that each road must appear in exactly one challenge, but a mountain may appear in several challenges.

Boboniu wants to minimize the total tiredness to do all the challenges. The tiredness of a challenge $M$ is the sum of tiredness of all mountains in it, i.e. $\sum_{i=1}^{|M|}t_{M_i}$.

He asked you to find the minimum total tiredness. As a reward for your work, you'll become a guardian in his Jianghu.

自从鲍伯牛建好他的江湖后，他每天都在这些山上练习功夫。

鲍伯牛为他的 $n$ 座山设计了一张地图。他用 $n-1$ 条道路将全部 $n$ 座山连通，使得任意两座山之间都可通过道路相互到达。

对于第 $i$ 座山，鲍伯牛估算了在其山顶练功的疲劳值 $t_i$；同时，他也估算了每座山的高度 $h_i$。

一条路径是一个山的序列 $M$，满足：对每个 $i$（$1 \le i < |M|$），$M_i$ 与 $M_{i+1}$ 之间存在一条道路。若对每个 $i$（$1 \le i < |M|$）均有 $h_{M_i} \le h_{M_{i+1}}$，则鲍伯牛将该路径视为一项挑战。

鲍伯牛希望将全部 $n-1$ 条道路划分成若干项挑战。注意：每条道路必须且仅能出现在一项挑战中，但一座山可以出现在多项挑战中。

鲍伯牛希望最小化完成所有挑战所需的总疲劳值。一项挑战 $M$ 的疲劳值定义为其中所有山的疲劳值之和，即 $\sum_{i=1}^{|M|}t_{M_i}$。

他请你求出最小总疲劳值。作为对你工作的嘉奖，你将成为他江湖中的一名守护者。

The first line contains a single integer $n$ ($2 \le n \le 2 \cdot 10^5$), denoting the number of the mountains.

The second line contains $n$ integers $t_1, t_2, \ldots, t_n$ ($1 \le t_i \le 10^6$), denoting the tiredness for Boboniu to do Kungfu on each mountain.

The third line contains $n$ integers $h_1, h_2, \ldots, h_n$ ($1 \le h_i \le 10^6$), denoting the height of each mountain.

Each of the following $n - 1$ lines contains two integers $u_i$, $v_i$ ($1 \le u_i, v_i \le n, u_i \neq v_i$), denoting the ends of the road. It's guaranteed that all mountains are connected via roads.

第一行包含一个整数 $n$（$2 \le n \le 2 \cdot 10^5$），表示山的数量。

第二行包含 $n$ 个整数 $t_1, t_2, \ldots, t_n$（$1 \le t_i \le 10^6$），表示 Boboniu 在每座山上练习功夫所消耗的疲劳值。

第三行包含 $n$ 个整数 $h_1, h_2, \ldots, h_n$（$1 \le h_i \le 10^6$），表示每座山的高度。

接下来的 $n - 1$ 行中，每行包含两个整数 $u_i$、$v_i$（$1 \le u_i, v_i \le n$，且 $u_i \neq v_i$），表示一条道路的两个端点。保证所有山通过道路相互连通。

Print one integer: the smallest sum of tiredness of all challenges.

输出一个整数：所有挑战的疲劳度总和的最小值。

For the first example:

In the picture, the lighter a point is, the higher the mountain it represents. One of the best divisions is:

• Challenge $1$: $3 \to 1 \to 2$
• Challenge $2$: $5 \to 2 \to 4$

The total tiredness of Boboniu is $(30 + 40 + 10) + (20 + 10 + 50) = 160$. It can be shown that this is the minimum total tiredness.

对于第一个样例：

图中，一个点越亮，代表其对应的山越高。其中一种最优划分方式为：

• 挑战 $1$：$3 \to 1 \to 2$
• 挑战 $2$：$5 \to 2 \to 4$

鲍伯牛的总疲劳值为 $(30 + 40 + 10) + (20 + 10 + 50) = 160$。可以证明这是最小的总疲劳值。