LeetCode 124 Binary Tree Maximum Path Sum - Core Idea

There are two values to track:

$1.$ Max path going down from $v$ (used for parent's calculation).

$2.$ Max path through $v$ (candidate for global answer).

Path through $v$ = $v + ext{leftMax} + ext{rightMax}$ (if both positive). Path from $v$ = $v + max( ext{leftMax}, ext{rightMax}, 0)$. Return this to parent. The parent can only extend one branch, not both. If the parent used both branches, it would create a branching structure, not a path.