  Recently it was shown that, if the subtree and chain reduction rules have
been applied exhaustively to two unrooted phylogenetic trees, the reduced trees
will have at most 15k-9 taxa where k is the TBR (Tree Bisection and
Reconnection) distance between the two trees, and that this bound is tight.
Here we propose five new reduction rules and show that these further reduce the
bound to 11k-9. The new rules combine the ``unrooted generator'' approach
introduced in [Kelk and Linz 2018] with a careful analysis of agreement forests
to identify (i) situations when chains of length 3 can be further shortened
without reducing the TBR distance, and (ii) situations when small subtrees can
be identified whose deletion is guaranteed to reduce the TBR distance by 1. To
the best of our knowledge these are the first reduction rules that strictly
enhance the reductive power of the subtree and chain reduction rules.
