  We study the conflict-free connection coloring of trees, which is also the
conflict-free coloring of the so-called edge-path hypergraphs of trees. We
first prove that for a tree $T$ of order $n$, $cfc(T)\geq cfc(P_n)=\lceil
\log_{2} n\rceil$, which completely confirms the conjecture of Li and Wu. We
then present a sharp upper bound for the conflict-free connection number of
trees by a simple algorithm. Furthermore, we show that the conflict-free
connection number of the binomial tree with $2^{k-1}$ vertices is $k-1$. At
last, we study trees which are $cfc$-critical, and prove that if a tree $T$ is
$cfc$-critical, then the conflict-free connection coloring of $T$ is equivalent
to the edge ranking of $T$.
