A new one-pass and tree-shaped tableau system for LTL sat- isfiability checking has been recently proposed, where each branch can be explored independently from others and, furthermore, directly cor- responds to a potential model of the formula. Despite its simplicity, it proved itself to be effective in practice. In this paper, we provide a SAT-based encoding of such a tableau system, based on the technique of bounded satisfiability checking. Starting with a single-node tableau, i.e., depth k of the tree-shaped tableau equal to zero, we proceed in an incremental fashion. At each iteration, the tableau rules are encoded in a Boolean formula, representing all branches of the tableau up to the current depth k. A typical downside of such bounded techniques is the effort needed to understand when to stop incrementing the bound, to guarantee the completeness of the procedure. In contrast, termination and completeness of the proposed algorithm is guaranteed without com- puting any upper bound to the length of candidate models, thanks to the Boolean encoding of the PRUNE rule of the original tableau system. We conclude the paper by describing a tool that implements our procedure, and comparing its performance with other state-of-the-art LTL solvers.