# 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.

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.
• 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