Is the Balanced Boolean Formula problem solvable in sublogarithmic space if the input has a tree structure?
It seems to me that the problem can be solved in constant space.

We perform a recursive procedure, and store only the parity of the number of negation gates seen on the path from the root to the current node.

At a leaf, if we have a variable and have seen an even number of negation gates, we set the variable to be true. Otherwise, if we have seen an odd number of negation gates, we set the variable to be false.

As before, at an AND or OR gate we first call recusively to the left, and then to the right if the value has not been determined.

Now if there exists a satisfying assignment, then the assignment defined above will satisfy the formula.