If turned sideways, so that the node labeled 1 is at the bottom, this may be regarded as the Lucas Tree, which grows according to the rules that
At the nth year there are Ln nodes.
Here is a different representation of the same tree.
This grows according to the rules that every mature branch sprouts a new branch at the end of each year, and new branches take a year to reach maturity, except that in the first year two new branches are formed instead of one.
Mature branches are indicated by heavy lines.
At the end of the nth year there are Ln branches.