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The number of embeddings of a partially ordered set $S$ in a partiallyordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both,$S$ and $T$, have only one unique maximal element, we define good embeddings asthose in which the maximal elements of $S$ and $T$ overlap. We investigate thenumber of good and all embeddings of a rooted poset $S$ in the family of allbinary trees on $n$ elements considering two cases: plane (when the order ofdescendants matters) and non-plane. Furthermore, we study the number ofembeddings of a rooted poset $S$ in the family of all planted plane trees ofsize $n$. We derive the asymptotic behaviour of good and all embeddings in allcases and we prove that the ratio of good embeddings to all is of the order$\Theta(1/\sqrt{n})$ in all cases, where we provide the exact constants.Furthermore, we show that this ratio is non-decreasing with $S$ in the planebinary case and asymptotically non-decreasing with $S$ in the non-plane binarycase and in the planted plane case. Finally, we comment on the case when $S$ isdisconnected.

Counting embeddings of rooted trees into families of rooted trees