  Given two non-empty graphs $H$ and $T$, write $H\succcurlyeq T$ to mean that
$t(H,G)^{|E(T)|}\geq t(T,G)^{|E(H)|}$ for every graph $G$, where
$t(\cdot,\cdot)$ is the homomorphism density function. We obtain various
necessary and sufficient conditions for two trees $H$ and $T$ to satisfy
$H\succcurlyeq T$ and determine all such pairs on at most 8 vertices. This
extends results of Leontovich and Sidorenko from the 1980s and 90s. Our
approach applies an information-theoretic technique of Kopparty and Rossman to
reduce the problem of showing that $H\succcurlyeq T$ for two forests $H$ and
$T$ to solving a particular linear program. We also characterize trees $H$
which satisfy $H\succcurlyeq S_k$ or $H\succcurlyeq P_4$, where $S_k$ is the
$k$-vertex star and $P_4$ is the $4$-vertex path.
