  We consider packing tree degree sequences in this paper. We set up a
conjecture that any arbitrary number of tree degree sequences without common
leaves have edge disjoint tree realizations. This conjecture is known to be
true for $2$ and $3$ tree degree sequences. In this paper, we give a proof for
$4$ tree degree sequences and a computer aided proof for $5$ tree degree
sequences. We also prove that for arbitrary $k$, $k$ tree degree sequences
without common leaves and at least $2k-4$ vertices which are not leaves in any
of the trees always have edge disjoint tree realizations. The main ingredient
in all of the presented proofs is to find rainbow matchings in certain
configurations.
