<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>

from collections import defaultdict
from typing import List
class Solution:
    def checkWays(self, connections: List[List[int]]) -> int:
        adjacency_list = defaultdict(list)
        node_degrees = defaultdict(int)
        edge_existence = defaultdict(lambda: defaultdict(bool))
        for start, end in connections:
            if end not in adjacency_list[start]:
                adjacency_list[start].append(end)
            if start not in adjacency_list[end]:
                adjacency_list[end].append(start)
            node_degrees[start] += 1
            node_degrees[end] += 1
            edge_existence[start][end] = True
            edge_existence[end][start] = True
        for node in adjacency_list:
            adjacency_list[node].sort(key=lambda x: -node_degrees[x])
        root_node = self.findRoot(node_degrees, len(adjacency_list))
        if root_node == -1:
            return 0
        self.hasAlternativeConnections = False
        if not self.depthFirstSearch(adjacency_list, root_node, node_degrees, edge_existence, set(), set()):
            return 0
        return 2 if self.hasAlternativeConnections else 1
    def findRoot(self, node_degrees: defaultdict(int), total_nodes: int) -> int:
        for node, degree in node_degrees.items():
            if degree == total_nodes - 1:
                return node
        return -1
    def depthFirstSearch(self, adjacency_list: defaultdict(list), current_node: int, node_degrees: defaultdict(int),
            edge_existence: defaultdict(lambda: defaultdict(bool)), visited_ancestors: set, visited_nodes: set) -> bool:
        visited_nodes.add(current_node)
        if not all(edge_existence[current_node][ancestor] for ancestor in visited_ancestors):
            return False
        visited_ancestors.add(current_node)
        for neighbor in adjacency_list[current_node]:
            if neighbor in visited_nodes:
                continue
            if node_degrees[neighbor] == node_degrees[current_node]:
                self.hasAlternativeConnections = True
            if not self.depthFirstSearch(adjacency_list, neighbor, node_degrees, edge_existence, visited_ancestors, visited_nodes):
                return False
        visited_ancestors.remove(current_node)
        return True

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