Branching form of the tree property

For an uncountable cardinal $\kappa$, this convention requires a cofinal branch in every separated rooted tree of height $\kappa$ with fewer than $\kappa$ immediate successors per node, without bounding level sizes. It is stronger than the usual tree property away from inaccessible cardinals. If $\mathrm{cf}(\kappa )<\kappa$, attach chains of cofinally increasing lengths below $\kappa$ above fewer than $\kappa$ children of one root; this contradicts the property. If $2^{\lambda }\ge \kappa$ for $\lambda <\kappa$, take a binary tree through level $\lambda$, choose $\kappa$ distinct nodes there, and attach to the $\xi$th a chain of length $\xi$. It has height $\kappa$, binary branching, and no cofinal branch. Hence the property implies that $\kappa$ is a strongly inaccessible cardinal. At such a cardinal a separated tree with small immediate branching has small levels, so this convention agrees with the usual tree property. It is not the different notion called the strong tree property in modern set theory.