The standard model Higgs sector, extended by one weak gauge triplet of scalar fields with a very small vacuum expectation value, is a very promising setting to account for neutrino masses through the so-called type II seesaw mechanism. In this paper we consider the general renormalizable doublet/triplet Higgs potential of this model. We perform a detailed study of its main dynamical features that depend on five dimensionless couplings and two mass parameters after spontaneous symmetry breaking, and highlight the implications for the Higgs phenomenology. In particular, we determine (i) the complete set of tree-level unitarity constraints on the couplings of the potential and (ii) the exact tree-level boundedness from below constraints on these couplings, valid for all directions. When combined, these constraints delineate precisely the theoretically allowed parameter space domain within our perturbative approximation. Among the seven physical Higgs states of this model, the mass of the lighter (heavier) CPeven state h(0) (H-0) will always satisfy a theoretical upper (lower) bound that is reached for a critical value mu(c) of mu (the mass parameter controlling triple couplings among the doublet/triplet Higgses). Saturating the unitarity bounds, we find an upper bound m(h)(0) < O(0.7-1 TeV), while the upper bound for the remaining Higgses lies in the range of several tens of TeV. However, the actual masses can be much lighter. We identify two regimes corresponding to mu greater than or similar to mu(c) and mu less than or similar to mu(c). In the first regime the Higgs sector is typically very heavy, and only h(0) that becomes SM-like could be accessible to the LHC. In contrast, in the second regime, somewhat overlooked in the literature, most of the Higgs sector is light. In particular, the heaviest state H-0 becomes SM-like, the lighter states being the CPodd Higgs, the (doubly) charged Higgses, and a decoupled h(0), possibly leading to a distinctive phenomenology at the colliders.