CF1188A2.Add on a Tree: Revolution

Note that this is the second problem of the two similar problems. You can hack this problem if you solve it. But you can hack the previous problem only if you solve both problems.

You are given a tree with $n$ nodes. In the beginning, $0$ is written on all edges. In one operation, you can choose any $2$ distinct leaves $u$, $v$ and any integer number $x$ and add $x$ to values written on all edges on the simple path between $u$ and $v$. Note that in previous subtask $x$ was allowed to be any real, here it has to be integer.

For example, on the picture below you can see the result of applying two operations to the graph: adding $2$ on the path from $7$ to $6$, and then adding $-1$ on the path from $4$ to $5$.

You are given some configuration of nonnegative integer pairwise different even numbers, written on the edges. For a given configuration determine if it is possible to achieve it with these operations, and, if it is possible, output the sequence of operations that leads to the given configuration. Constraints on the operations are listed in the output format section.

Leave is a node of a tree of degree $1$. Simple path is a path that doesn't contain any node twice.

注意，这是两个相似问题中的第二个问题。如果你解决了本题，就可以对本题进行 Hack；但只有当你同时解决了这两个问题时，才能对前一个问题进行 Hack。

你被给定一棵包含 $n$ 个节点的树。初始时，所有边上均写有数字 $0$。在一次操作中，你可以任选两个不同的叶子节点 $u$、$v$，以及任意一个整数 $x$，并将 $x$ 加到 $u$ 与 $v$ 之间简单路径上的所有边所写的数值上。注意：在前一子任务中，$x$ 可以是任意实数；而在此题中，$x$ 必须为整数。

例如，在下图中，展示了对图执行两次操作后的结果：第一次操作将 $2$ 加到从节点 $7$ 到节点 $6$ 的路径上，第二次操作将 $-1$ 加到从节点 $4$ 到节点 $5$ 的路径上。

你被给定一种边上的配置：每条边上写有一个非负偶数，且所有这些数两两不同。对于该给定配置，请判断是否能通过上述操作达到它；若可以，请输出一组达成该配置的操作序列。关于操作的约束条件详见“输出格式”部分。

叶子节点（Leaf）是指度数为 $1$ 的树节点。简单路径（Simple path）是指不重复经过任何节点的路径。

The first line contains a single integer $n$ ($2 \le n \le 1000$) — the number of nodes in a tree.

Each of the next $n-1$ lines contains three integers $u$, $v$, $val$ ($1 \le u, v \le n$, $u \neq v$, $0 \le val \le 10\,000$), meaning that there is an edge between nodes $u$ and $v$ with $val$ written on it. It is guaranteed that these edges form a tree. It is guaranteed that all $val$ numbers are pairwise different and even.

第一行包含一个整数 $n$（$2 \le n \le 1000$）—— 表示树中节点的数量。

接下来的 $n-1$ 行，每行包含三个整数 $u$、$v$、$val$（$1 \le u, v \le n$，$u \neq v$，$0 \le val \le 10\,000$），表示节点 $u$ 与节点 $v$ 之间存在一条边，且该边上写有数值 $val$。保证这些边构成一棵树。保证所有 $val$ 两两不同且均为偶数。

If there aren't any sequences of operations which lead to the given configuration, output "NO".

If it exists, output "YES" in the first line. In the second line output $m$ — number of operations you are going to apply ($0 \le m \le 10^5$). Note that you don't have to minimize the number of the operations!

In the next $m$ lines output the operations in the following format:

$u, v, x$ ($1 \le u, v \le n$, $u \not = v$, $x$ — integer, $-10^9 \le x \le 10^9$), where $u, v$ — leaves, $x$ — number we are adding.

It is guaranteed that if there exists a sequence of operations producing given configuration, then there exists a sequence of operations producing given configuration, satisfying all the conditions above.

如果不存在任何操作序列能够得到给定的构型，则输出 “NO”。

如果存在这样的操作序列，则第一行输出 “YES”；第二行输出 $m$ —— 即你将执行的操作次数（$0 \le m \le 10^5$）。注意：你不必最小化操作次数！

接下来的 $m$ 行中，按如下格式输出各操作：

$u, v, x$（其中 $1 \le u, v \le n$，$u \ne v$，$x$ 为整数，且 $-10^9 \le x \le 10^9$），这里 $u, v$ 为叶子节点，$x$ 为要加上的数值。

题目保证：若存在某个操作序列能生成给定构型，则必存在一个满足上述所有条件的操作序列。

The configuration from the first sample is drawn below, and it is impossible to achieve.

The sequence of operations from the second sample is illustrated below.

第一个样例的配置如下图所示，且无法达成。

第二个样例的操作序列如下图所示。