A tree-shellable function is a positive Boolean function which can be represented by a binary decision tree whose number of paths from the root to a leaf labeled 1 equals the number of prime implicants. In this paper, we consider the tree-shellability of DNFs with restrictions. We show that, for read-k DNFs, the number of terms in a tree-shellable function is at most k². We also show that, for k-DNFs, recognition of ordered tree-shellable functions is NP-complete for k=4 and tree-shellable functions can be recognized in polynomial time for constant k.