Binary tree node value maximization
This problem is known as Independent Set (and not two-coloring). Your intuition is quite correct though.

As a hint, I'd like to offer the following definitions:

Let $n$ be a node of the binary tree. Let $P(n)$ be the maximum profit that can be attained from the subtree rooted at $n$ given that $n$ itself may not be included. Similarly, let $Q(n)$ be the maximum profit that can be attained from the subtree rooted at $n$ given that $n$ itself may be included.