  Given a finite set $X$, a collection $\mathcal{T}$ of rooted phylogenetic
trees on $X$ and an integer $k$, the Hybridization Number problem asks if there
exists a phylogenetic network on $X$ that displays all trees from $\mathcal{T}$
and has reticulation number at most $k$. We show two kernelization algorithms
for Hybridization Number, with kernel sizes $4k(5k)^t$ and $20k^2(\Delta^+-1)$
respectively, with $t$ the number of input trees and $\Delta^+$ their maximum
outdegree. Experiments on simulated data demonstrate the practical relevance of
these kernelization algorithms. In addition, we present an $n^{f(k)}t$-time
algorithm, with $n=|X|$ and $f$ some computable function of $k$.
