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.

翻译：给定两个非空图$H$和$T$，记$H\succcurlyeq T$表示对任意图$G$均有$t(H,G)^{|E(T)|}\geq t(T,G)^{|E(H)|}$，其中$t(\cdot,\cdot)$为同态密度函数。我们得到了两棵树$H$和$T$满足$H\succcurlyeq T$的各种必要条件和充分条件，并确定了顶点数不超过8的所有此类树对。这一结果推广了Leontovich和Sidorenko在20世纪80年代和90年代的成果。我们的方法应用了Kopparty和Rossman的信息论技术，将证明两个森林$H$和$T$满足$H\succcurlyeq T$的问题简化为求解特定线性规划。我们还刻画了满足$H\succcurlyeq S_k$或$H\succcurlyeq P_4$的树$H$，其中$S_k$是$k$个顶点的星图，$P_4$是4个顶点的路径图。