  Using the foundations laid down in Hardy and Harris (2006) ["A new
formulation of the spine approach in branching diffusions",
arXiv:math.PR/0611054], we present new spine proofs of the L^p-convergence
p>=1) of some key `additive' martingales for three distinct models of branching
diffusions, including new results for a multi-type branching Brownian motion
and discussion of left-most particle speeds. The spine techniques we develop
give clear and simple arguments in the spirit of the conceptual spine proofs
found in Kyprianou (2004) ["Travelling wave solutions to the KPP equation, Ann.
Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] and Lyons et al (1997)
["A conceptual proof of the Kesten-Stigum theorem for multi-type branching
processes", Classical and modern branching processes, IMA Vol. Math. Appl.,
vol.84, Springer, New York, pp181-185], and they should also extend to more
general classes of branching diffusions. Importantly, the techniques in this
paper also pave the way for the path large-deviation results for branching
diffusions found in Hardy and Harris (2006) ["A conceptual approach to a path
result for branching Brownian motion", Stochastic Processes and their
Applications, doi:10.1016/j.spa.2006.05.010].
