The Average Height of Catalan Trees by Counting Lattice Paths

The average height of Catalan trees of a given size is a structural parameter important in the analysis of algorithms, as it measures the expected maximum cost of a search in a tree. This parameter has been studied first with generating functions and complex variable theory, yielding an asymptotic approximation. Later on, real analysis was used instead of complex analysis. We have further reduced the conceptual difficulty by replacing generating functions with the enumeration of monotonic lattice paths, whose graphical representations make the derivation much more intuitive.