 Successor-Suslin decomposition

For an infinite cardinal number $\kappa$, every ${\kappa }^{+}$-Suslin set is a union of ${\kappa }^{+}$ many $\kappa$-Suslin sets. A countable sequence of ordinals below the successor cardinal ${\kappa }^{+}$ is bounded there, so restrict the representing tree successively to labels below each $\alpha <{\kappa }^{+}$.