  We derive asymptotics for the moments as well as the weak limit of the height
distribution of watermelons with p branches with wall. This generalises a
famous result of de Bruijn, Knuth and Rice on the average height of planted
plane trees, and results by Fulmek and Katori et al. on the expected value,
respectively the higher moments, of the height distribution of watermelons with
two branches.
  The asymptotics for the moments depend on the analytic behaviour of certain
multidimensional Dirichlet series. In order to obtain this information we prove
a reciprocity relation satisfied by the derivatives of one of Jacobi's theta
functions, which generalises the well known reciprocity law for Jacobi's theta
functions.
