I have n sectors, enumerated 0 to n-1 counterclockwise. The boundaries between these sectors are infinite branches (n of them).


These branches meet at certain points (junctions). Each junction is adjacent to a subset of the sectors (at least 3 of them). 


By specifying what sectors my junctions are adjacent to, I can completely recover the tree.
This seems like something known, but I would like a reference to it.


The number of trees with n branches is given by 
<http://www.oeis.org/A001003>
and this is quite easy to prove. 


Furthermore, if I order the sectors in the description of the junctions, I can make this representation unique.


Example:
(0,1,2,3,4,5) represents the tree with only one vertex, and 6 branches connected to this junction.



Describe a tree by junctions
I realized that taking the dual of my trees, I always get an n-gon,
where some chords, the faces in the dual are my junctions. The bijection is now trivial.


