  We give an $n^{O(\log\log n)}$-time membership query algorithm for properly
and agnostically learning decision trees under the uniform distribution over
$\{\pm 1\}^n$. Even in the realizable setting, the previous fastest runtime was
$n^{O(\log n)}$, a consequence of a classic algorithm of Ehrenfeucht and
Haussler. Our algorithm shares similarities with practical heuristics for
learning decision trees, which we augment with additional ideas to circumvent
known lower bounds against these heuristics. To analyze our algorithm, we prove
a new structural result for decision trees that strengthens a theorem of
O'Donnell, Saks, Schramm, and Servedio. While the OSSS theorem says that every
decision tree has an influential variable, we show how every decision tree can
be "pruned" so that every variable in the resulting tree is influential.
