I don't know much about tree representations, but looking at your list I haven't seen the (binary) tree as an expression in algebra:
$$\langle A, \bot, (\bullet, \bullet) \rangle, $$
where $A$ is the set of all expressions that could be generated from this algebra. For example, a leaf $\bot$, the smallest tree $ (\bot, \bot)$, a tree of height one $((\bot,\bot),\bot)$. Often the leaf $\bot$ is denoted as $()$ and what you get is just parentheses expressions generated by simple CFG grammar (in here you get collection of $n$-ary trees, e.g. $(()()())(()()()())$ would be two trees of degree 3 and 4):
$$ T \to T(T) \mid \varepsilon.$$
If you need more than the structure, then the binary tree could be denoted as (e.g. from Haskell):
BTree α = Leaf | Node (BTree α) α (BTree α)
where $\alpha$ is some type parameter. When you get bored, you can make it into a generating function:
\begin{align}
T[\alpha] &= 1 + T[\alpha] \times \alpha \times T[\alpha] \\
0 &= \alpha \times T^2[\alpha] - T[\alpha] + 1 \\
T_1[\alpha] &= \frac{1-\sqrt{1-4\alpha}}{2\alpha} \\
T_2[\alpha] &= \frac{1+\sqrt{1-4\alpha}}{2\alpha}
\end{align}
The first solution $T_1[\alpha]$ coincides with the generating function of Catalan numbers which not by chance describes the number of binary trees of size $n$. So $\frac{1-\sqrt{1-4\alpha}}{2\alpha}$ (the second root has a pole at zero) could be expanded into a infinite sum $\sum_{n=0}^{\infty}c_n\alpha^n$ and we gain one more representation of (binary) trees as the following:
$$T[\alpha] = \coprod_{n=0}^{\infty} c_n\alpha^n$$
where $c_n$ are the Catalan numbers: 1, 1, 2, 5, 14, 42, etc.
Finally, if you don't like binary trees, you could use left-son, right-brother trick or play with some $n$-ary tree representation like this one:
Tree α = Node α (List (Tree α))
List β = Nil | Cons β (List β)
I hope it helps ;-)
