  A normal network is uniquely determined by the set of phylogenetic trees that
it displays. Given a set $\mathcal{P}$ of rooted binary phylogenetic trees,
this paper presents a polynomial-time algorithm that reconstructs the unique
binary normal network whose set of displayed binary trees is $\mathcal{P}$, if
such a network exists. Additionally, we show that any two rooted phylogenetic
trees can be displayed by a normal network and show that this result does not
extend to more than two trees. This is in contrast to tree-child networks where
it has been previously shown that any collection of rooted phylogenetic trees
can be displayed by a tree-child network. Lastly, we introduce a type of
cherry-picking sequence that characterises when a collection $\mathcal{P}$ of
rooted phylogenetic trees can be displayed by a normal network and, further,
characterise the minimum number of reticulations needed over all normal
networks that display $\mathcal{P}$. We then exploit these sequences to show
that, for all $n\ge 3$, there exist two rooted binary phylogenetic trees on $n$
leaves that can be displayed by a tree-child network with a single
reticulation, but cannot be displayed by a normal network with less than $n-2$
reticulations.
