We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et al.(2011). Proving an exponential lower bound for the size of the non-deterministic thrifty branching programs would separate NL from P under the thrifty hypothesis. In this context, we consider a restriction of non-deterministic thrifty branching programs called bitwise-independence. We show that any bitwise-independent non-deterministic thrifty branching program solving tree evaluation problems on trees of height h and k-ary values for nodes, must have at least 21kh2 states. Prior to this work, lower bounds were known for general branching programs only for fixed heights h=234 (Cook et al., 2011). Our lower bounds are also tight (up to a factor of k), since the known(Cook et al., 2011) non-deterministic thrifty branching programs for this problem of size O(kh2+1) are bitwise-independent. We prove our results by associating a fractional pebbling strategy with any bitwise-independent non-deterministic thrifty branching program solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights.

Our main technique is the entropy method introduced by Jukna and Zak(2001) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known (Cook et al., 2011) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any k-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.