Fast embedding of spanning trees in biased Maker-Breaker games

Given a tree T = ( V , E ) on n vertices, we consider the ( 1 : q ) Maker-Breaker tree embedding game T n . The board of this game is the edge set of the complete graph on n vertices. Maker wins T n if and only if he is able to claim all edges of a copy of T. We prove that there exist real numbers α , ε > 0 such that, for sufficiently large n and for every tree T on n vertices with maximum degree at most n ε , Maker has a winning strategy for the ( 1 : q ) game T n , for every q ⩽ n α . Moreover, we prove that Maker can win this game within n + o ( n ) moves which is clearly asymptotically optimal.