A characterization of the tree $T^*$ such that $\mathrm{BP}(T^*)=\overleftrightarrow{\mathrm{DFUDS}(T)}$, the reversal of $\mathrm{DFUDS}(T)$ is given. An immediate consequence is a rigorous characterization of the tree $\hat{T}$ such that $\mathrm{BP}(\hat{T})=\mathrm{DFUDS}(T)$. In summary, $\mathrm{BP}$ and $\mathrm{DFUDS}$ are unified within an encompassing framework, which might have the potential to imply future simplifications with regard to queries in $\mathrm{BP}$ and/or $\mathrm{DFUDS}$. Immediate benefits displayed here are to identify so far unnoted commonalities in most recent work on the Range Minimum Query problem, and to provide improvements for the Minimum Length Interval Query problem.