We construct a single Lindström quantifier Q such that FO(Q), the extension of first-order logic with Q has the same expressive power as monadic second-order logic on the class of binary trees (with distinct left and right successors) and also on unranked trees with a sibling order. This resolves a conjecture by ten Cate and Segoufin. The quantifier Q is a variation of a quantifier expressing the Boolean satisfiability problem.
Dawar, A., & Segoufin, L. (2015). Capturing MSO with one quantifier. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9300, pp. 142–152). Springer Verlag. https://doi.org/10.1007/978-3-319-23534-9_8