Given the root of a binary tree of integers, return the maximum path sum.

To find the maximum path sum in a binary tree, we can use a recursive approach. For each node, we calculate the maximum path sum that starts at that node and goes down into one of its subtrees (left or right). This is done by taking the node's value and adding the maximum of 0 and the maximum path sum from its children. We also consider the path that passes through the current node, which includes the node's value plus the maximum path sums from both its left and right children. We keep track of the overall maximum path sum encountered globally during the traversal. The base case for the recursion is when we encounter a null node, in which case the path sum is 0.