Rooted NNI moves and distance-1 tail moves on tree-based phylogenetic networks

Any two tree-based rooted binary phylogenetic networks (with the same number of reticulation nodes and identical degrees at the root) are connected by at most $(4k+8)\cdot\left(|X|+\frac34k\right)+|X|\cdot\lceil\log_2|X|\rceil$ rooted nearest-neighbor interchange (rNNI) moves. As a corollary of the proof we obtain that the diameter is at most a $2k$ additive term larger for distance-1 tail moves.