<p>You are given an array <code>pairs</code>, where <code>pairs[i] = [x<sub>i</sub>, y<sub>i</sub>]</code>, and:</p>

<ul>
	<li>There are no duplicates.</li>
	<li><code>x<sub>i</sub> &lt; y<sub>i</sub></code></li>
</ul>

<p>Let <code>ways</code> be the number of rooted trees that satisfy the following conditions:</p>

<ul>
	<li>The tree consists of nodes whose values appeared in <code>pairs</code>.</li>
	<li>A pair <code>[x<sub>i</sub>, y<sub>i</sub>]</code> exists in <code>pairs</code> <strong>if and only if</strong> <code>x<sub>i</sub></code> is an ancestor of <code>y<sub>i</sub></code> or <code>y<sub>i</sub></code> is an ancestor of <code>x<sub>i</sub></code>.</li>
	<li><strong>Note:</strong> the tree does not have to be a binary tree.</li>
</ul>

<p>Two ways are considered to be different if there is at least one node that has different parents in both ways.</p>

<p>Return:</p>

<ul>
	<li><code>0</code> if <code>ways == 0</code></li>
	<li><code>1</code> if <code>ways == 1</code></li>
	<li><code>2</code> if <code>ways &gt; 1</code></li>
</ul>

<p>A <strong>rooted tree</strong> is a tree that has a single root node, and all edges are oriented to be outgoing from the root.</p>

<p>An <strong>ancestor</strong> of a node is any node on the path from the root to that node (excluding the node itself). The root has no ancestors.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img src="https://assets.leetcode.com/uploads/2020/12/03/trees2.png" style="width: 208px; height: 221px;" />
<pre>
<strong>Input:</strong> pairs = [[1,2],[2,3]]
<strong>Output:</strong> 1
<strong>Explanation:</strong> There is exactly one valid rooted tree, which is shown in the above figure.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/12/03/tree.png" style="width: 234px; height: 241px;" />
<pre>
<strong>Input:</strong> pairs = [[1,2],[2,3],[1,3]]
<strong>Output:</strong> 2
<strong>Explanation:</strong> There are multiple valid rooted trees. Three of them are shown in the above figures.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> pairs = [[1,2],[2,3],[2,4],[1,5]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There are no valid rooted trees.</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= pairs.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= x<sub>i </sub>&lt; y<sub>i</sub> &lt;= 500</code></li>
	<li>The elements in <code>pairs</code> are unique.</li>
</ul>

class Solution:
    def checkWays(self, pairs: List[List[int]]) -> int:
        
        graph = defaultdict(set)
        for node1, node2 in pairs:
            graph[node1].add(node2)
            graph[node2].add(node1)

        pq = []
        for node in graph:
            heapq.heappush(pq, (-len(graph[node]), node))
        
        N = len(pq)
        visited = set()
        multiple = False
        while pq:
            size, node = heapq.heappop(pq)
            size = -size

            root = 0
            root_size = N + 1
            if len(visited) > 0:
                for node_next in graph[node]:
                    if len(graph[node_next]) < root_size and node_next in visited:
                        root = node_next
                        root_size = len(graph[node_next])

            visited.add(node)
            if root == 0:
                if size != N - 1:
                    return 0
                continue

            for node_next in graph[node]:
                if node_next == root:
                    continue
                if node_next not in graph[root]:
                    return 0
            if root_size == size:
                multiple = True
        if multiple:
            return 2
        return 1
        
    #     return self.helper(set(graph.keys()), graph)
    # def helper(self, nodes, graph):
    #     degree = defaultdict(list)
    #     total = len(nodes) - 1

    #     for node in nodes:
    #         num_connected = len(graph[node])
    #         degree[num_connected].append(node)
        
    #     if len(degree[total]) == 0:
    #         return 0
        
    #     root = degree[total][0]
    #     for node in graph[root]:
    #         graph[node].remove(root)
        
    #     visited = set()
    #     i = 0
    #     components = defaultdict(set)
    #     def dfs(node, i):
    #         components[i].add(node)
    #         visited.add(node)
    #         for node_next in graph[node]:
    #             if node_next not in visited:
    #                 dfs(node_next, i)
        
    #     for node in nodes:
    #         if node != root and node in visited:
    #             dfs(node, i)
    #             i += 1

    #     candidates = [self.helper(components[i]) for i in comps]
    #     if 0 in candidates:
    #         return 0
    #     if 2 in candidates or len(degree[total]) >= 2:
    #         return 2
    #     return 1
48500