(my failed proof)
The Split Theorem
The merge and sort trick
Efficient polynomial algorithm?
Introduced the RNNI graph on ranked trees (to the best of our knowledge)
Established basic geometric properties of the graph
Proved that all the fancy NNI methods, e.g. Sleator-Tarjan-Thurston and merge-and-sort argument, don't work
Failed to prove that RNNI is polynomial-time decidable
Why is this hard?
Trees are many!