We consider two critical spatial branching processes on
$\mathbb{R}^d$: critical branching Brownian motion, and the critical
Dawson-Watanabe process. A basic feature of these processes is that their
ergodic behavior is highly dimension dependent. It is known that in low
dimensions, $d \leq 2$, the only invariant measure is $\delta_0$ , the unit
point mass on the empty state. In high dimensions, $d \geq 3$, there is a
family ${\nu_{\theta}, \theta \epsilon [0, \infty)}$ of extremal invariant
measures; the measures ${\nu_{\theta}$ are translation invariant and indexed by
spatial intensity. We prove here, for $d \geq 3$, that all invariant measures
are convex combinations of these measures.