  We use the algorithm of Cole et al. (2000) to generate merger trees for the
first star clusters in a Lambda CDM cosmology under an isotropic UV background
radiation field, parametrized by J_21. We have investigated the problem in two
ways: a global radiation background and local radiative feedback surrounding
the first star clusters.
  Cooling in the first halos at high redshift is dominated by molecular
hydrogen, H_2 - we call these Generation 1 objects. At lower redshift and
higher virial temperature, T_vir > 10^4K, electron cooling dominates - we call
these generation 2. Radiation fields act to photo-dissociate H_2, but also
generate free electrons that can help to catalyse its production. At modest
radiation levels, J_{21}/(1+z)^3 ~ 10^{-12}-10^{-7}, the nett effect is to
enhance the formation of Generation 1 star-clusters. At higher fluxes the
heating from photo-ionisation dominates and halts their production. With a
realistic build-up of flux over time, the period of enhanced H_2 cooling is so
fleeting as to be barely discernable and the nett effect is to move primordial
star cluster formation towards Generation 2 objects at lower redshift. A
similar effect is seen with local feedback. Provided that enough photons are
produced to maintain ionization of their host halo, they will suppress the
cooling in Generation 1 halos and boost the numbers of primordial star clusters
in Generation 2 halos. Significant suppression of Generation~1 halos occurs for
specific photon fluxes in excess of about 10^{43} ph s^{-1} Msun^{-1}.
