In this work we define a metric structure to compare functions defined on different merge trees. The metric introduced possesses some stability properties, which we illustrate within a standard topological data analysis (TDA) framework, and can be computed with a dynamical binary linear programming approach. We showcase the effectiveness of the whole framework with simulated data sets. Using functions defined on merge trees proves to be very effective in situations where other topological data analysis tools, like persistence diagrams, cannot be used meaningfully.