Ranked alphabet is very often used in Ranked Trees definition, like here for instance. In that example for given set $\Sigma=\{a,b,c\}$ ranks assigned by arity function $ar : \Sigma\rightarrow\mathcal{N}$ as:
$ar(a)=2, ar(b)=2, ar(c)=1$.
And Ranked Tree over $\Sigma$ is defined as:
$T_{\Sigma_r}$, the set of ranked trees, is the smallest set of terms $f(t_1,\dots,t_k)$ such that: $f\in\Sigma_r$, $k = ar(f)$, and $t_i\in T_{\Sigma_r}$ for all $1\leq i\leq k$.
The tree in this example looks like:
b
/ \
a b
/ \ / \
b c c c
/ \
c c
But what about trees like that?
b
/ \
a b
/ \ / \
b c c c
| |
c a
This is also valid tree, but it is obviously is unranked.
My question: do any research regarding unranked alphabet trees exist?
What I've found so far is related only to logic for unranked trees.
