Vaught's Conjecture for Sums of Products of Rooted Trees

We confirm Vaught's conjecture for each partial order ${\mathbb X}=\sum _{\mathbb I}\prod _{j<m_i}{\mathbb X}_{i,j}$ from the closure $\langle \langle {\mathcal C}^{\rm rt}\rangle _Π\rangle _Σ$ of the class ${\mathcal C}^{\rm rt}$ of rooted trees under finite direct products and lexicographic sums.