Summary: In this paper, we introduce the concept of split and non-split tree \((D, C)\)-set of a connected graph \(G\) and its associated color variable, namely split tree \((D, C)\) number and non-split tree \((D, C)\) number of \(G\). A subset \(S\subseteq V\) of vertices in \(G\) is said to be a split tree \((D, C)\) set of \(G\) if \(S\) is a tree \((D, C)\) set and \(\langle V - S\rangle\) is disconnected. The minimum size of the split tree \((D, C)\) set of \(G\) is the split tree \((D, C)\) number of \(G\), \(\gamma_{\chi_{ST}}(G) = \min\{|S| : S\text{ is a split tree }(D, C)\text{ set}\}\). A subset \(S \subseteq V\) of vertices of \(G\) is said to be a non-split tree \((D, C)\) set of \(G\) if \(S\) is a tree \((D, C)\) set and \(\langle V - S\rangle\) is connected and non-split tree \((D, C)\) number of \(G\) is \(\gamma_{\chi_{NST}}(G) = \min\{|S| : S\text{ is a non-split tree }(D, C)\text{ set of }G\}\). The split and non-split tree \((D, C)\) number of some standard graphs and its compliments are identified.