Balancedness of MSO transductions in polynomial time

We show that MSO definable tree-to-string transductions with a fixed copy number, given as top-down tree-to-string transducers with fixed copying-bound, can be checked for balancedness of all output strings in polynomial time. We also present an upper bound for the copying bound for deterministic top-down transducers and show that in general, balancedness of output languages of deterministic finite-copying top-down tree-to-string transducers is DExptime-hard.