In the following, I consider rooted, unlabelled, ordered binary trees, where each node has exactly $0$ or $2$ children (I will simply call them binary trees). A binary tree $t'$ is a subtree of a binary tree $t$ if there exists a node $v$ in $t$ such that the tree rooted at this node $v$ equals $t'$.
For a binary tree $t$, let the size $|t|$ be the number of leaves of $t$, i.e. the number of nodes that have zero children.
I am interested in the following property of a binary tree $t$: If two subtrees $t'$ and $t''$ of $t$ have the same size, i.e. $|t'|=|t''|$, then the subtrees have to be equal, i.e. $t'=t''$. In other words, the number of different subtrees of $t$ that have size $m$ is at most one for each $m$.
Question: What I want to know is the number of different trees of size $n$ fulfilling this property?
I tried some of the usual combinatorial approaches but I made no progress so far.