Introduction To Question
Context: Universal Algebra
I
Definition: A $\mathtt{S}$-algebra is an algebra $\langle A, succ, \bullet \rangle, $ with one unary operation and no identities.
Let $\mathsf{S}(X)$ be the free $\mathtt{S}$-algebra over the finite set $X$. Let $\mathrm{N} = \mathtt{S}({\emptyset})$.
Immediately we have elements $x \in N$ such as
$\bullet$,
$S(\bullet)$,
$S(S(\bullet))$,
$S(S(S(\bullet)))$ and so on...
These elements may be labelled "zero", "one", "two", "three", and etc as they are named in $\mathbb{N}$.
II
Definition: A $\mathtt{B}$-algebra is an algebra $\langle A, \circ, \bullet\rangle$ with one binary operation, one nullary operation and no identities.
Let $\mathsf{T}(X)$ be the free $\mathtt{T}$-algebra over the finite set $X$. Let $\mathtt{B} = \mathtt{T}({\emptyset})$. Immediately we have elements $x \in \mathtt{B}$ such as
$\bullet$,
$\bullet \circ \bullet$,
$(\bullet \circ \bullet) \circ \bullet$,
$\bullet \circ (\bullet \circ \bullet)$,
$(\bullet \circ \bullet) \circ (\bullet \circ \bullet)$,
and so on...
These elements do not have common labels that I am aware of.
Question
Does $\mathtt{B}$ have an existing notation? Do the elements $x \in \mathtt{B}$ have common names like the elements of $\mathtt{N}$ (and $\mathbb{N}$) have?
I prefer the notation $\mathbb{B}$ (for binary), partially because it seems appropriate given the parallelism with the definition $\mathtt{N}$ and the associated natural numbers $\mathbb{N}$. However blackboard both letters are usually reserved for the "big players" $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ with multiplicative operations. Also the notation $\mathbb{N}$ usually denotes the full structure of the natural numbers complete with addition and multiplication, not just the underlying set together with a successor operation and a constant.
https://en.wikipedia.org/wiki/Blackboard_bold
https://web.cecs.pdx.edu/~sheard/course/Cs163/Doc/FullvsComplete.html
I would not recommend $\Bbb B$, which is already commonly used for the 2-element Boolean algebra (or for the corresponding discrete space in topology).
As explained in the second given link, your algebra is the algebra of full binary trees. Thus, if you really need a notation, you may try something like ${\Bbb T}_2$ (T for tree and 2 for binary).