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 Tn. The board of this game is the edge set of the complete graph on n vertices. Maker wins Tn 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 nε, Maker has a winning strategy for the (1:q) game Tn, for every q⩽nα. Moreover, we prove that Maker can win this game within n+o(n) moves which is clearly asymptotically optimal.