What's the number of possible structures of alkanes $C_n H_{2n+2}$?

Question

When my chemistry teacher started listing out all possible structure of the hydrocarbon $C_7H_{16}$, my mind flied to look for a general formula. Let me mathematicalize this problem.

Here, we have $n$ points on a plane, all connected with some lines, following these rules:

1. no points can have more than $4$ lines attached;

2. number of lines between any $2$ points is $1$;

3. no loop formed.

We only concern how the points are linked by lines, their positions on the plane doesn't matter. So what is the number of possible structure when there's $n$ points?

E.g., when there's $5$ points, all possibilities are: (I use $C$ to denote points)

When there's 6,

I started by finding an algorithm to list out all possibilities, I found this too hard and turned to find a way to check if 2 structure are actually the same, which I also failed later. I suspect this has something to do with recursive things. Please help, Thanks.

p.s. I'm not sure if I've made the rules clearly, please point out any problem you see.

Answer 1

Here is a start for what you are looking for. The number of different carbon chains is the same as the number of non isomorphic trees.