# Formal Mathematical Terminology For Tree Diagrams

I currently have a tree diagram that shows the probabilities for certain paths in a game. The tree diagram first branches into four possibilities and then another four possibilities for each of the beginning four, making a total of sixteen possibilities (numbered outcome 1 - 16). I would like to know the proper mathematical notation or term to address the "root" of the branches that lead down to outcome 4 - 8. I have called the starting point of the whole tree diagram the "root" so I feel like it is confusing to term the branches of outcomes 4 - 8 that merge as another "root". Currently I am referring to its probability as "P(Outcome 4|8)" but is that an acceptable way to refer to it?

Edit: Now that I learned the definitions, my problem is that I do not know how to call the node on the 2nd level that branches out into leafs (outcomes which are on the third level) 4 to 8.

Trees are actually mathematically well-defined objects and have various pleasant properties and definitions associated with them. What you're talking about is probably better called a Directed Acyclic Graph, which is slightly different from a tree, but not in any important respect for this problem. A sketch of your tree would help me out quite a lot, but here's a picture of one that we can talk about to fix some definitions. The circles are each referred to as nodes or vertices. The lines between them are edges, and the edge between nodes 10 and 5 is labelled or weighted with $x$. Each of the nodes at the bottom - 5, 18, 36, 39, 49 - is called a leaf. The node at the top - 33 - is called the root. Each stratum of nodes in the tree is called a level, though conventions for numbering levels differ. The level containing only 33 is either the $1^{st}$ or the $0^th$ level. You say that the branches of outcomes 4-8 "merge", presumably into a single node. This kind of node doesn't really have a name, but you can name nodes as you like.

For clarifying your problem, I'd advise naming each level of your tree with a random variable - $X_0$ at the root, then $X_1$, then $X_2$ at the leaves. The leaves could then have labels like $O_1, O_2, ...$, and we could talk about $\mathbb{P}(X_2 = O_4 | X_1 = 8)$, for example.

A sketch of your graph and a better statement of your problem would let me edit this into a better answer, but this will probably get you started on the terminology.

• This is perfect. Just one thing: what do you mean when you say X2=O4|X1=8? My tree diagram is basically like yours expect that there are four edges branching out of the root and then four edges branching out from all the nodes in the 2nd level, for a total of 16 leafs. So how should I refer to the node that is on the 2nd level and is second to the left? Would that be O2? Oct 18, 2013 at 12:13
• Actually, does it make sense to call that "the node that branches to O(subscript)4-O(subscript)8"? Oct 18, 2013 at 12:19
• The vertical bar means "given", so $\mathbb{P}(X_2 = O_4 | X_1 = 8)$ is the probability that $X_2$ takes the value $O_4$, given that we already know $X_1=8$. The probability that the next person I meet is taller than 180cm is some value. The probability that the next person I meet is taller than 180cm given that I know they'll be a man is a somewhat different value. In your question you used the vertical bar notation which did confuse me a little. Oct 19, 2013 at 8:58
• As for what to call things, writing long english names is usually discouraged, and it is generally preferred to draw out a diagram of your tree, label everything with a short name, and go from there. You might consider $N_{i,j}$ for the $i^{th}$ node on the $j^{th}$ level, or vice versa, or you might want to call your leaves $O_1, ..., O_{16}$ and the interior nodes $N_1, ..., N_5$. The correct choice is whatever makes your point most clearly to your audience. Oct 19, 2013 at 9:00