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.