Herman Haupt (1817-1905), was instrumental in moving the process of designing bridges from art to science. Earlier, the design process was largely empirical and trial and error. He wanted formulas to calculate the strength. Herman asked the advice of prominent engineers and only Benjamin Latrobe had ever attempted such calculations. There were no existing formulas for complex trusses.
Herman built models and developed his own formulas - which lead to his 1839 patent for an "improved lattice truss". It consisted of many verticals and multiple intersecting diagonals, all sloped toward the bridge center. It does not include an arch. Few were built, and according to historian J.G. James, Herman "converted" to using the Howe truss.
Many of the bridges that claim to be Haupt truss have an arch. And some of them don't have multiple intersecting diagonals like the patent describes.
And then there is the cast iron, wrought iron bridges categorized as Haupt trusses. These are a Pratt truss overlayed with an arch.
Bridgehunter has examples of each of these in this category.