Enumeration of monadic second-order queries on trees

We consider the enumeration problem of monadic second-order (MSO) queries with first-order free variables over trees. In [Bagan 2006] it was shown that this problem is in CONSTANT-DELAYlin. An enumeration problem belongs to CONSTANT-DELAYlin if for an input structure of size n it can be solved by: —an O(n) precomputation phase building an index structure, —followed by a phase enumerating the answers with no repetition and a constant delay between two consecutive outputs. In this article we give a different proof of this result based on the deterministic factorization forest decomposition theorem of Colcombet [Colcombet 2007]