Counting the number of tree when the set of the subtrees is given
You can't do it at all, let alone do it efficiently. Consider a tree&nbsp;$T$ and let its set of subtrees be&nbsp;$S$. If you're told $B=S$, the cardinality could be anything between $1$ and&nbsp;$|S|$, inclusive, because $A$&nbsp;could be any subset of&nbsp;$S$ that contains&nbsp;$T$.