Tree formalisms The intuitive notion of a tree in mathematics is quite straightforward. However, there are several different formalisms of the tree concept. The link http://ncatlab.org/nlab/show/tree lists several possibilities. I would like to have an exhaustive list of all existing tree formalisms. Any references to such a source will be much appreciated. Otherwise, any tree formalism other than those appearing in the link will be appreciated. 
I'm particularly interested in comparing the resulting categories of trees and noting where a particular formalism is useful and, if possible, point to the particular aspect of the formalism that makes it suitable for a particular application. So hopefully each formalism of trees will also have a notino of morphism of trees yielding a category. Different definitions of tree can yield radically different categories. Thanks!
 A: 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 ;-)
