The Kneser chromatic function distinguishes trees

R.P. Stanley defined a invariant for graphs called the chromatic symmetric function and conjectured it is complete invariant for trees. Miezaki et al. generalised the chromatic symmetric function and defined the Kneser chromatic functions denoted by $\{X_{K_{\mathbb{N},k}}\}_{k\in\mathbb{N}}$, and rephrase Stanley's conjecture that $X_{K_{\mathbb{N},1}}$ is a complete invariant for trees. This paper shows $X_{K_{\mathbb{N},2}}$ is a complete invariant for trees.