CF1637F.Towers

You are given a tree with $n$ vertices numbered from $1$ to $n$. The height of the $i$-th vertex is $h_i$. You can place any number of towers into vertices, for each tower you can choose which vertex to put it in, as well as choose its efficiency. Setting up a tower with efficiency $e$ costs $e$ coins, where $e \gt 0$.

It is considered that a vertex $x$ gets a signal if for some pair of towers at the vertices $u$ and $v$ ($u \neq v$, but it is allowed that $x = u$ or $x = v$) with efficiencies $e_u$ and $e_v$, respectively, it is satisfied that $\min(e_u, e_v) \geq h_x$ and $x$ lies on the path between $u$ and $v$.

Find the minimum number of coins required to set up towers so that you can get a signal at all vertices.

给你一棵包含 $n$ 个顶点的树，顶点编号为 $1$ 到 $n$。第 $i$ 个顶点的高度为 $h_i$。你可以在任意多个顶点上放置信号塔；对于每座塔，你可以自由选择其放置的顶点以及其效率值。设置一座效率为 $e$ 的塔需要花费 $e$ 枚金币，其中 $e > 0$。

称顶点 $x$ 接收到信号，当且仅当存在两座位于顶点 $u$ 和 $v$（$u \neq v$；但允许 $x = u$ 或 $x = v$）的塔，其效率分别为 $e_u$ 和 $e_v$，满足：
$\min(e_u, e_v) \geq h_x$，且 $x$ 位于 $u$ 与 $v$ 之间的唯一路径上。

求使所有顶点均能接收到信号所需的最少金币数。

The first line contains an integer $n$ ($2 \le n \le 200\,000$) — the number of vertices in the tree.

The second line contains $n$ integers $h_i$ ($1 \le h_i \le 10^9$) — the heights of the vertices.

Each of the next $n - 1$ lines contain a pair of numbers $v_i, u_i$ ($1 \le v_i, u_i \le n$) — an edge of the tree. It is guaranteed that the given edges form a tree.

第一行包含一个整数 $n$（$2 \le n \le 200\,000$）—— 树中顶点的数量。

第二行包含 $n$ 个整数 $h_i$（$1 \le h_i \le 10^9$）—— 各顶点的高度。

接下来的 $n - 1$ 行，每行包含一对数字 $v_i, u_i$（$1 \le v_i, u_i \le n$）—— 树的一条边。保证所给的边构成一棵树。

Print one integer — the minimum required number of coins.

输出一个整数——所需的最少硬币数量。

In the first test case it's optimal to install two towers with efficiencies $2$ at vertices $1$ and $3$.

In the second test case it's optimal to install a tower with efficiency $1$ at vertex $1$ and two towers with efficiencies $3$ at vertices $2$ and $5$.

In the third test case it's optimal to install two towers with efficiencies $6$ at vertices $1$ and $2$.

在第一个测试用例中，最优方案是在顶点 $1$ 和 $3$ 处各安装一座效率为 $2$ 的塔。

在第二个测试用例中，最优方案是在顶点 $1$ 处安装一座效率为 $1$ 的塔，并在顶点 $2$ 和 $5$ 处各安装一座效率为 $3$ 的塔。

在第三个测试用例中，最优方案是在顶点 $1$ 和 $2$ 处各安装一座效率为 $6$ 的塔。