We consider biased (1:b) Avoider-Enforcer games in the monotone and strict
versions. In particular, we show that Avoider can keep his graph being a
forest for every but maybe the last round of the game if b≥200nlnn. By this we
obtain essentially optimal upper bounds on the threshold biases for the non-
planarity game, the non-k-colorability game, and the Kt-minor game thus
addressing a question and improving the results of Hefetz, Krivelevich,
Stojaković, and Szabó. Moreover, we give a slight improvement for the lower
bound in the non-planarity game.