Dual properties of monotonically normal spaces and generalized trees

In the first part of this note, we show that every monotonically normal space is dually scattered of rank ≤2.In the second part of this note, we introduce a notion of a generalized tree with the Sorgenfrey topology (a generalized Sorgenfrey topology). A partially ordered set X is said to be a generalized tree   if (←,x)={y∈X:y