On the Complexity of a Game Related to the Dictionary Problem

A game on trees that is related to the dictionary problem is considered. There are two players, A and B, which take turns. Player A models the user of the dictionary and player B models the implementation of it. At his turn, 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, which forces a cost of $\Omega (n \log \log n)$ for an n-game, i.e., a game in which each player takes n turns, and that there is a strategy for player B, which keeps the cost within $O(n \log \log n)$.