  In this paper we consider positional games where the winning sets are tree
universal graphs. Specifically, we show that in the unbiased Maker-Breaker game
on the complete graph $K_n$, Maker has a strategy to occupy a graph which
contains copies of all spanning trees with maximum degree at most $cn/\log(n)$,
for a suitable constant $c$ and $n$ being large enough. We also prove an
analogous result for Waiter-Client games. Both of our results show that the
building player can play at least as good as suggested by the random graph
intuition. Moreover, they improve on a special case of earlier results by
Johannsen, Krivelevich, and Samotij as well as Han and Yang for Maker-Breaker
games.
