  Recently, harmonic functions and frequently universal harmonic functions on a
tree $T$ have been studied, taking values on a separable Fr\'{e}chet space $E$
over the field $\mathbb{C}$ or $\mathbb{R}$. In the present paper, we allow the
functions to take values in a vector space $E$ over a rather general field
$\mathbb{F}$. The metric of the separable topological vector space $E$ is
translation invariant and instead of harmonic functions we can also study more
general functions defined by linear combinations with coefficients in
$\mathbb{F}$. Unlike the past literature, we don't assume that $E$ is complete
and therefore we present a new argument, avoiding Baire's theorem.
