We study a certain family of typed branching diffusions where the type of each particle moves as an Ornstein-Uhlenbeck process and binary branching occurs at a rate quadratic in the particle's type. We calculate the ‘left-most' particle speed for the branching process explicitly, aided by close connections with harmonic oscillator theory. The behaviour of the system changes markedly below a certain critical temperature parameter. In the high-temperature regime, the study of various ‘additive' martingales and their use in a change of measure method provides the proof of the almost sure speed of spread of the particle system. Also, we briefly mention how to use the martingale results of the branching diffusion model in representations of travelling-wave solutions for the associated reaction-diffusion equation.