Universal functions and $\Sigma_{\omega}$-bounded structures

We introduce the notion of a $\Sigma_{\omega}$-bounded structure and specify a necessary and sufficient condition for a universal $\Sigma$-function to exist in a hereditarily finite superstructure over such a structure, for the class of all unary partial $\Sigma$-functions assuming values in the set $\omega$ of natural ordinals. Trees and equivalences are exemplified in hereditarily finite superstructures over which there exists no universal $\Sigma$-function for the class of all unary partial $\Sigma$-functions, but there exists a universal $\Sigma$-function for the class of all unary partial $\Sigma$-functions assuming values in the set $\omega$ of natural ordinals. We construct a tree $T$ of height $5$ such that the hereditarily finite superstructure ${\mathbb {HF}}(T)$ over $T$ has no universal $\Sigma$-function for the class of all unary partial $\Sigma$-functions assuming values $0, 1$ only.