The Stacks project

Definition 15.106.6. Let $A$ be a local ring with henselization $A^ h$ and strict henselization $A^{sh}$. The number of branches of $A$ is the number of minimal primes of $A^ h$ if finite and $\infty$ otherwise. The number of geometric branches of $A$ is the number of minimal primes of $A^{sh}$ if finite and $\infty$ otherwise.