Trees with three branch vertices are determined by their chromatic symmetric functions

Every finite simple tree with exactly three vertices of degree at least three is determined, among all finite simple graphs, by its chromatic symmetric function. The proof combines a power-sum derivative of an alternating trunk deletion sum with the path sequence and the three-leaf part of the subtree polynomial. The result closes the exact three-branch-vertex layer. It does not resolve Stanley's conjecture for all trees or the layer with four or more branch vertices. Status: Internally reviewed preprint; external mathematical review and formal peer review are pending.