The tree search game for two players

We consider a two-player search game on a tree T. One vertex (unknown to the players) is randomly selected as the target. The players alternately guess vertices. If a guess v is not the target, then both players are informed in which subtree of T \ v the target lies. The winner is the player who guesses the target. When both players play optimally, we show that each of them wins with probability approximately 1/2. When one player plays optimally and the other plays randomly, we show that the player with the optimal strategy wins with probability between 9/16 and 2/3 (asymptotically). When both players play randomly, we show that each wins with probability between 13/30 and 17/30 (asymptotically).