This question evaluates understanding of binary tree structures, computation of path-based metrics (diameter), handling edge cases such as empty or single-node trees, and the ability to analyze time and space complexity.

Given the root of a binary tree, compute the tree's diameter defined as the maximum number of edges on any path between two nodes. Return the diameter, explain whether paths must pass through the root, and provide a solution with time and space complexity analysis. Handle empty and single-node trees.