Let $T$ be a tree, ${\mathop{\rm End}\nolimits}(T)$ be the number of ends of $T$ and let $L(T)$ be the infimum of topological entropies of transitive maps of $T$. We give an elementary approach to the estimate that $L(T)\ge (1/{\mathop{\rm End}\nolimits} (T))\log 2$. We also divide the set of all trees (up to homeomorphisms) into pairwise disjoint subsets ${\cal P}(i)$, $i\in \{0\}\cup\N$ and prove that $L(T)=(1/({\mathop{\rm End}\nolimits} (T)-i))\log 2$ if $T\in {\cal P}(i)$ with $i=0,1$, and $L(T)\le (\text{respectively} =) (1/({\mathop{\rm End}\nolimits} (T)-i))\log 2$ if $T\in {\cal P}(i)$ (respectively $T\in {\cal P}'(i))$ with $i\ge 2$, where ${\cal P}'(i)$ is an infinite subset of ${\cal P}(i)$. Furthermore, we show that there is a tree $T$ such that the topological entropy of each transitive map of $T$ is larger than $L(T)$, and hence disprove a conjecture of Alseda et al (1997).