We present an adaptive tree-log scheme to improve the performance of checking the integrity of arbitrarily-large untrusted data, when using only a small fixed-sized trusted state. Currently, hash trees are used to check the data. In many systems that use hash trees, programs perform many data operations before performing a critical operation that exports a result outside of the program's execution environment. The adaptive tree-log scheme we present uses this observation to harness the power of the constant runtime bandwidth overhead of a log-based scheme. For all programs, the adaptive tree-log scheme's bandwidth overhead is guaranteed to never be worse than a parameterizable worst case bound. Furthermore, for all programs, as the average number of times the program accesses data between critical operations increases, the adaptive tree-log scheme's bandwidth overhead moves from a logarithmic to a constant bandwidth overhead.