Dynamic membership for any fixed regular tree language is maintained in O(log n / log log n) per relabeling, with an exact constant-time class under a standard hardness conjecture.
Marked ancestor problems http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.71.3834&rep=rep1&type=pdf
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Dynamic Membership for Regular Tree Languages
Dynamic membership for any fixed regular tree language is maintained in O(log n / log log n) per relabeling, with an exact constant-time class under a standard hardness conjecture.