# What's the name of this type of tree?

What's the name of a tree that looks like the below?

I've provided a few iterations $d = 0,1, \dotsc$, because it looks sort of fractal. The 0's are nodes, and the lines, edges.

$d=0$

0


$d=1$

  0
|
0-0-0
|
0


$d = 2$

       0
|
0  0-0-0  0
|    |    |
0-0----0----0-0
|    |    |
0  0-0-0  0
|
0


I was thinking of calling them "lattice trees", but after a bit of Googling, it looks like these might be called "star-like" trees, but if anyone could give a reference or a confirmation that would be nice.

The way I've constructed them is by setting the length of the maximum path in the graph to be $2d$ and then tried to have as many vertices as possible.

• If you order the tree from the nodes of $d$ up-down, setting the center node at the top, what is doing is just adding three new nodes from each node of the previous level (except for $d=0$ where the tree add four new nodes). – Masacroso Apr 21 '17 at 6:13
• If we were to consider the center vertex as the root, if it weren't for the fact that it had four children, I would have called it a full, complete and balanced ternary tree, or a perfect ternary tree. Every interior vertex has three children except that one. – JMoravitz Apr 21 '17 at 6:15
• These are certainly not star-like trees, because the Wikipedia definition is violated by your last graph. There are more than one vertices with degree greater than $2$ in it. – Shraddheya Shendre Apr 21 '17 at 6:15
• @ShraddheyaShendre Ah good to know that option has been eliminated! :) – jamesh625 Apr 21 '17 at 6:38
• They look like the balls of radius $d$ about the identity, in the Cayley graph of the free group on 2 generators. The wiki page for the free group has this as its first image. You could equivalently consider these as balls of radius $d$ in the infinite 4-valent tree. – Joppy Apr 21 '17 at 6:58