I know almost nothing in combinatorics, so this question might be very easy, or well-known.
Fix a number $n$. We will consider rooted planar binary trees with $n$ leaves. We will distinguish between the root (some fixed vertex of valency $1$), leaves (vertices of valency $1$ that are not the root) and internal vertices (vertices of valency $3$).
On these trees we can define the following operations. For any internal vertex $v$ of a tree $T$ we can flip the two branches growing up from $v$ as it is shown on the picture:
Then we say two trees $T_1$ and $T_2$ are equivalent if one can be obtained from the other by some combination of the operations described above.
So my question is: how can we describe the set of equivalences classes? I would be happy with the answer of the type: we can (easily) encode trees into some "data", and then each equivalence class is represented by the "data" that satisfies some conditions. Here "data" is maybe a sequence of numbers, or something else (I don't know combinatorics, so I don't know what are the efficient tools to deal with this kind of problem).
Thank you very much for your help!