In this paper, we consider one-to-one encodings for a discrete memoryless source, which are ?one-shot? encodings associating a distinct codeword with each source symbol. Such encodings could be employed when only a single source symbol rather than a sequence of source symbols needs to be transmitted. We consider two slightly different types of one-to-one encodings depending on whether the empty codeword is used or not. Given that the probability of the most likely source symbol is available, we provide several new lower and upper bounds on the expected length of optimal one-to-one codes.