Boolean--Eulerian numbers

We study decreasing binary trees in which every vertex with two children is colored red or blue. We construct two bijections. The first, to ordered set partitions into odd-sized blocks each arranged as an alternating permutation, shows that the exponential generating function of these trees is $1\mathrm{/}(1-\tan z)$. The second, to nonplane decreasing 1-2 trees paired with a binary label on each non-root vertex, proves combinatorially that the count equals $2^{n-1}$ times the~$n$th Euler number. Refining by the number of right edges yields the Boolean–Eulerian polynomials, which are an explicit algebraic transform of the classical Eulerian polynomials. The Foata–Strehl orbit decomposition, recast in the decreasing-binary-tree model, gives a direct combinatorial proof of gamma-positivity, and the algebraic transform carries real-rootedness and interlacing of zeros from the Eulerian polynomials to the Boolean–Eulerian polynomials.