A question about binary trees

I need help to identify where does appear the following algebra of trees.

UPDATE: seems that it does not appear anywhere else. Thanks for input, even if it led to this negative result. Please let me know, though, if you recognize this algebra somewhere!

We start with the set of binary trees:

  • the root I is a tree
  • if A and B are trees then AB is the tree which is obtained from A and B by adding o the root I the LEFT child A and the RIGHT child B

We think therefore about oriented binary trees, so that any node which is not a leaf has two childs, called the left and the right child.

On the set of these binary trees we define two operations:

  • a 1-ary operation denoted by a *
  • a 2-ary operation denoted by a little circle

The operations are defined recursively, as in the following picture:

question

I am very interested to learn about the appearance of this algebra somewhere. My guess is that this algebra has been studied. For the moment I don’t have any information about this and I kindly request for help.

If not, what can be said about it? It is easy to see that the root is a neutral element for the operation “small circle”. Probably the operation “small circle” is associative, however this is less clear than I first thought.

If you think this structure is too dry to be interesting, then just try it, take examples and see what gives. How much does it take to compute the result of an operation, etc…

Thank you for help, if any!

 

 

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