Summary: We present two simpler versions of the $\log(n)$ time parallel recognition of unambiguous context-free languages on CREW PRAM’s. The first algorithm combines the ideas of the algorithms given in {\it K.-J. Lange} and {\it P. Rossmanith} [Lect. Notes Comput. Sci. 452, 399-406 (1990; Zbl 0731.68038)] and {\it W. Rytter} [Inf. Comput. 73, 75-86 (1987; Zbl 0633.68075)]. The result related to that algorithm is an unambiguous decomposition of trees into smaller ones. The analysis of this algorithm is quite simple. The second algorithm is based on unique paths decomposing trees into smaller subtrees. It has a shorter description and it is simpler to explain how it works. However, its correctness is nontrivial. Both algorithms reduce the number of processors by a linear factor in the case of general unambiguous cfl’s. For linear unambiguous cfl’s the number of processors is reduced to $O(n\sp 2)$ by a very simple algorithm. We show how to remove write conflicts in parallel computations on trees by using two kinds of unique decompositions of trees. Finally we show that our results can be generalized for finitely ambiguous cfl’s.