We develop a model of regular, infinite hypertrees, to mimic for hypergraphs what infinite trees do for graphs. We then examine two notions of spectra or "first eigenvalue" for the infinite tree, obtaining a precise value for the first notion and obtaining some estimates for the second. The results indicate agreement of the first eigenvalue of the infinite hypertree with the "second eigenvalue" of a random hypergraph of the same degree, to within logarithmic factors, at least for the first notion of first eigenvalue.