A recently proposed balanced-bracket encoding (Yli-Jyrä and GómezRodríguez 2017) has given us a way to embed all noncrossing dependency graphs into the string space and to formulate their exact arcfactored inference problem (Kuhlmann and Johnsson 2015) as the best string problem in a dynamically constructed and weighted unambiguous context-free grammar. The current work improves the encoding and makes it shallower by omitting redundant brackets from it. The streamlined encoding gives rise to a bounded-depth subset approximation that is represented by a small finite-state automaton. When bounded to 7 levels of balanced brackets, the automaton has 762 states and represents a strict superset of more than 99.9999% of the noncrossing trees available in Universal Dependencies 2.4 (Nivre et al. 2019). In addition, it strictly contains all 15-vertex noncrossing digraphs. When bounded to 4 levels and 90 states, the automaton still captures 99.2% of all noncrossing trees in the reference dataset. The approach is flexible and extensible towards unrestricted graphs, and it suggests tight finite-state bounds for dependency parsing, and for the main existing parsing methods.