Fluorescent spectroscopy experiments with single-enzyme molecules yield a large volume of statistical data that can be analyzed and interpreted using stochastic models of enzyme action. Here, we present two models, each based on the mechanism that an enzyme molecule must pass through a sequence of conformational transformations to complete its catalytic turnover cycle. in the simplest model, only one path leading to the release of product is present. In contrast to this, two different catalytic paths are possible in the second considered model. If a cycle is started from an active state, immediately after the previous product release, it follows a different conformational route and is much shorter. Our numerical investigations show that both models generate nonMarkovian molecular statistics. However, their memory landscapes and distributions of cycle times are significantly different. The memory landscape of the double-path model bears strong similarity to the recent experimental data for horseradish peroxidase.