Valid Tree Determination

You are tasked with determining if a given set of edges represents a valid tree structure. A valid tree is a fundamental concept in graph theory and computer science, appearing in data structures, algorithms, and network design. Understanding how to identify a valid tree is crucial for many applications.

Problem Description

Given the number of nodes n and a list of undirected edges, edges, determine if the graph represented by these edges is a valid tree.

A graph is a valid tree if and only if the following two conditions are met:

1. Connectivity: All nodes in the graph must be connected. There should be a path between any two nodes.
2. Acyclicity: The graph must not contain any cycles. Each node should have exactly one parent (except for the root, which has none), and there should be no way to return to a previously visited node without traversing back along the same edge.

In other words, a valid tree with n nodes must have exactly n-1 edges and be connected.

Input:

• n: An integer representing the total number of nodes. Nodes are labeled from 0 to n-1.
• edges: A list of lists, where each inner list [u, v] represents an undirected edge between node u and node v.

Output:

• TRUE if the graph is a valid tree.
• FALSE otherwise.

Edge Cases:

• A graph with a single node (n = 1) and no edges is a valid tree.
• A graph with n nodes and fewer than n-1 edges cannot be connected, hence not a tree.
• A graph with n nodes and more than n-1 edges must contain a cycle, hence not a tree.

Examples

Example 1:

Input: n = 5, edges = [[0, 1], [0, 2], [0, 3], [1, 4]]
Output: TRUE
Explanation: The graph is connected and has no cycles. It forms a valid tree.

Example 2:

Input: n = 5, edges = [[0, 1], [1, 2], [2, 3], [1, 3], [1, 4]]
Output: FALSE
Explanation: The graph contains a cycle (1 -> 2 -> 3 -> 1).

Example 3:

Input: n = 4, edges = [[0, 1], [2, 3]]
Output: FALSE
Explanation: The graph is not connected. There are two separate components.

Example 4:

Input: n = 1, edges = []
Output: TRUE
Explanation: A single node with no edges is considered a valid tree.

Constraints

• 1 <= n <= 2000
• 0 <= edges.length <= 5000
• edges[i].length == 2
• 0 <= u, v < n
• u != v
• There are no duplicate edges in edges.
• The graph might not be connected.

Notes

• You can use graph traversal algorithms like Breadth-First Search (BFS) or Depth-First Search (DFS) to check for connectivity and cycles.
• Alternatively, a Union-Find data structure can be very efficient for this problem, especially for detecting cycles and tracking connected components.
• Consider the initial check: if the number of edges is not equal to n-1, it cannot be a tree. However, this is not a sufficient condition on its own. You still need to verify connectivity and acyclicity.