They might mean best-case vs average-case vs worst-case. A worst-case hash table insertion might be O(n) (rehashing the table) and in the malicious case it might even be O(n^2) (all table entries collide in a single linked list).
mabbo probably means that the average insert time is O(1), but s/he's wrong and couldn't possibly be right. An O(1) average insertion time into a binary search (i.e., ordered) tree would mean O(n) to build the tree from n elements, which would mean an O(n) search ... no can do. And the paper clearly says that the O(1) restructuring is in addition to the O(lg n) time to find the insertion point ... which anyone familiar with BST algorithms would expect.