The number of permutations of length 2n1 avoiding 231 in the classical sense which can be realized as labels on an increasing binary tree read in the order they appear in a breadthfirst search. (Note that breadthfirst search reading word is equivalent to reading the tree left to right by levels, starting with the root.)
In some cases, more than one tree results in the same breadthfirst search reading word, but here we count the permutations, not the trees.
