We maintain the maximum spanning tree of a planar point set, as points are inserted or deleted, in O(log3 n) expected time per update in Mulmuley's average-case model of dynamic geometric computation. We use as subroutines dynamic algorithms for two other geometric graphs: the farthest neighbor forest and the rotating caliper graph related to an algorithm for static computation of point set widths and diameters. We maintain the former graph in expected time O(log2 n) per update and the latter in expected time O(log n) per update. We also use the rotating caliper graph to maintain the diameter, width, and minimum enclosing rectangle of a point set in expected time O(log n) per update. A subproblem uses a technique for average-case orthogonal range search that may also be of interest.
Eppstein, D. (1996). Average case analysis of dynamic geometric optimization. Computational Geometry: Theory and Applications, 6(1), 45–68. https://doi.org/10.1016/0925-7721(95)00018-6