Embeddings and other mappings of rooted trees into complete trees

Let $T^n$ be the complete binary tree of height $n$, with root $1_n$ as the maximum element. For $T$ a tree, define $A(n;T) = |\{S \subseteq T^n : 1_n \in S, S \cong T\}|$ and $C(n;T) = |\{S \subseteq T^n : S \cong T\}|$. We disprove a conjecture of Kubicki, Lehel and Morayne, which claims that $\frac{A(n;T_1)}{C(n;T_1)} \leq \frac{A(n;T_2)}{C(n;T_2)}$ for any fixed $n$ and arbitrary rooted trees $T_1 \subseteq T_2$. We show that $A(n;T)$ is of the form $\sum_{j=0}^l q_j(n) 2^{jn}$ where $l$ is the number of leaves of $T$, and each $q_j$ is a polynomial. We provide an algorithm for calculating the two leading terms of $q_l$ for any tree $T$. We investigate the asymptotic behaviour of the ratio $A(n;T)/C(n;T)$ and give examples of classes of pairs of trees $T_1, T_2$ where it is possible to compare $A(n;T_1)/C(n;T_1)$ and $A(n;T_2)/C(n;T_2)$. By calculating these ratios for a particular class of pairs of trees, we show that the conjecture fails for these trees, for all sufficiently large $n$. Kubicki, Lehel and Morayne have proved the conjecture when $T_1, T_2$ are restricted to being binary trees. We also look at embeddings into other complete trees, and we show how the result can be viewed as one of many possible correlation inequalities for embeddings of binary trees. We also show that if we consider strict order-preserving maps of $T_1,T_2$ into $T^n$ (rather than embeddings) then the corresponding correlation inequalities for these maps also generalise to arbitrary trees.