  In phylogenetics, a central problem is to infer the evolutionary
relationships between a set of species $X$; these relationships are often
depicted via a phylogenetic tree -- a tree having its leaves univocally labeled
by elements of $X$ and without degree-2 nodes -- called the "species tree". One
common approach for reconstructing a species tree consists in first
constructing several phylogenetic trees from primary data (e.g. DNA sequences
originating from some species in $X$), and then constructing a single
phylogenetic tree maximizing the "concordance" with the input trees. The
so-obtained tree is our estimation of the species tree and, when the input
trees are defined on overlapping -- but not identical -- sets of labels, is
called "supertree". In this paper, we focus on two problems that are central
when combining phylogenetic trees into a supertree: the compatibility and the
strict compatibility problems for unrooted phylogenetic trees. These problems
are strongly related, respectively, to the notions of "containing as a minor"
and "containing as a topological minor" in the graph community. Both problems
are known to be fixed-parameter tractable in the number of input trees $k$, by
using their expressibility in Monadic Second Order Logic and a reduction to
graphs of bounded treewidth. Motivated by the fact that the dependency on $k$
of these algorithms is prohibitively large, we give the first explicit dynamic
programming algorithms for solving these problems, both running in time
$2^{O(k^2)} \cdot n$, where $n$ is the total size of the input.
