Upper and Lower Bounds on $T_1$ and $T_2$ Decision Tree Model

We study a decision tree model in which one is allowed to query subsets of variables. This model is a generalization of the standard decision tree model. For example, the $\vee -$decision (or $T_1$-decision) model has two queries, one is a bit-query and one is the $\vee$-query with arbitrary variables. We show that a monotone property graph, i.e. nontree graph is lower bounded by $n\log n$ in $T_1$-decision tree model. Also, in a different but stronger model, $T_2$-decision tree model, we show that the majority function and symmetric function can be queried in $\frac{3n}{4}$ and $n$, respectively.