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Journal Articles (Research Report) The Electronic Journal of Combinatorics Year : 2022

Counting embeddings of rooted trees into families of rooted trees

Abstract

The number of embeddings of a partially ordered set $S$ in a partially ordered 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 as those in which the maximal elements of $S$ and $T$ overlap. We investigate the number of good and all embeddings of a rooted poset $S$ in the family of all binary trees on $n$ elements considering two cases: plane (when the order of descendants matters) and non-plane. Furthermore, we study the number of embeddings of a rooted poset $S$ in the family of all planted plane trees of size $n$. We derive the asymptotic behaviour of good and all embeddings in all cases 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 plane binary case and asymptotically non-decreasing with $S$ in the non-plane binary case and in the planted plane case. Finally, we comment on the case when $S$ is disconnected.
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Dates and versions

hal-03462575 , version 1 (03-12-2021)

Identifiers

Cite

Bernhard Gittenberger, Zbigniew Gołębiewski, Isabella Larcher, Małgorzata Sulkowska. Counting embeddings of rooted trees into families of rooted trees. The Electronic Journal of Combinatorics, 2022, 29 (3), pp.1-34. ⟨10.37236/10760⟩. ⟨hal-03462575⟩
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