Counting Embeddings of Rooted Trees into Families of Rooted Trees

Abstract

An embedding of a rooted tree $S$ into another rooted tree $T$ is a mapping of the vertex set of $S$ into the vertex set of $T$ that is order-preserving in a certain sense, depending on the chosen tree family. Equivalently, $S$ and $T$ may be interpreted as tree-like partially ordered sets, where $S$ is isomorphic to a subposets of $T$. A good embedding is an embedding where the root of $S$ is mapped to the root of $T$. We investigate the number of good and the number of all embeddings of a rooted tree $S$ into the family of all trees of given size $n$ of a certain family of trees. The tree families considered are binary plane trees (the order of descendants matters), binary non-plane trees and rooted plane trees. We derive the asymptotic behaviour of the number of good and the number of all embeddings in all these cases and prove that the ratio of the number of good embeddings to that of all embeddings is of the order $\Theta(1/\sqrt{n})$ in all cases, where we provide the exact constants as well. Furthermore, we prove some monotonicity properties of this ratio. Finally, we comment on the case where $S$ is disconnected.