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 ${\mathcal T}_n$. The board of this game is the edge set of the complete graph on $n$ vertices. Maker wins ${\mathcal T}_n$ if and only if he is able to claim all edges of a copy of $T$. We prove that there exist real numbers $\alpha, \epsilon > 0$ such that, for sufficiently large $n$ and for every tree $T$ on $n$ vertices with maximum degree at most $n^{\epsilon}$, Maker has a winning strategy for the $(1 : q)$ game ${\mathcal T}_n$, for every $q \leq n^{\alpha}$. Moreover, we prove that Maker can win this game within $n + o(n)$ moves which is clearly asymptotically optimal.