Recoverability for three or more source trees
Characterize recoverability from k-buckets when the bucket is formed from three or more rooted binary phylogenetic trees, beginning with the case of three source trees and extending to general numbers of source trees.
References
A further natural question, of course, is how the behaviour changes as we modify the number of trees, which we shall denote $m$. Are the characterisations notably different if $m=3$?
If this proves tractable, an interesting direction would be asymptotics, as $m$ approaches infinity. It seems likely that for fixed $n$, as $m$ increases recoverability becomes more difficult, but does the proportion of recoverable trees become vanishingly small, or approach a fixed proportion?
Similar to the question of the phase transition from infinite to finite counterexamples when modifying $k$, if our supposition that increasing $m$ makes recoverability increasingly difficult, one can also ask at what point does recoverability become less likely than unrecoverability? Is there an $m$ at which the finite family of counterexamples for $(n-1)$-buckets becomes an infinite family?
In particular, it would be striking if one could show a minimum required number of subtrees required to recover our main trees.
For instance, what is the algorithmic complexity of recovering the set of input trees given a $k$-bucket in general?
Secondly, we did not characterise the image of $k$-buckets in the present manuscript, and any future research would certainly benefit if there was a characterisation beyond something as simple as ``these subtrees can be partitioned into $m$ sets of compatible subtrees''.