On the complexity of a game related to the dictionary problem

A game on trees, which is related to the dictionary problem is considered. There are two players A and B who take turns. Player A models the user of the dictionary, and player B models its implementation. Player A modifies the tree by adding new leaves, and player B modifies the tree by replacing subtrees. The cost of an insertion is the depth of the new leaf, and the cost of an update is the size of the subtree replaced. The goal of player A is to maximize cost, and the goal of B is to minimize it. It is shown that there is a strategy for player A that forces a cost of Ω(n log log n ) for an n-game, that is, a game consisting of n turns of both players, and a strategy for player B that keeps the cost in O(n log log n)