  We present the first polynomial time algorithm to learn nontrivial classes of
languages of infinite trees. Specifically, our algorithm uses membership and
equivalence queries to learn classes of $\omega$-tree languages derived from
weak regular $\omega$-word languages in polynomial time. The method is a
general polynomial time reduction of learning a class of derived $\omega$-tree
languages to learning the underlying class of $\omega$-word languages, for any
class of $\omega$-word languages recognized by a deterministic B\"{u}chi
acceptor. Our reduction, combined with the polynomial time learning algorithm
of Maler and Pnueli [1995] for the class of weak regular $\omega$-word
languages yields the main result. We also show that subset queries that return
counterexamples can be implemented in polynomial time using subset queries that
return no counterexamples for deterministic or non-deterministic finite word
acceptors, and deterministic or non-deterministic B\"{u}chi $\omega$-word
acceptors.
  A previous claim of an algorithm to learn regular $\omega$-trees due to
Jayasrirani, Begam and Thomas [2008] is unfortunately incorrect, as shown in
Angluin [2016].
