A note on D-spaces and L-special trees

Let β∈ω1 be a limit ordinal and let Iα be a Dedekind complete linearly ordered metric space for each α∈β such that maxIα and minIα exist. If L=∏α∈βIα is under the lexicographic ordering such that any Lα-special tree is hereditarily a D-space for each α∈β, where Lα=∏γ∈αIγ is under the lexicographic ordering, then every L-special tree T satisfies that the height of each branch of T is countable. As a corollary, we get that if T is a [0,1]ω2-special tree, where [0,1]ω2 is under the lexicographic ordering, then the height of each branch of T is countable.