Combining Paths from Children

To maximize a path through node $v$, pick the two best downward paths from different children. If $v$ has children with max downward paths $p_1, p_2, l\dots, p_k$, the best path through $v$ is $v + p_1 + p_2$ where $p_1$ and $p_2$ are the two largest.

You cannot use the same child twice. A path visits each node at most once. This constraint forces you to pick two different child subtrees. In a binary tree, you have at most two children, making the choice simple.