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Let $\mathrm{cof}(\mu)=\mu$ and $\kappa$ be a supercompact cardinal with$\mu<\kappa$. Assume that there is an increasing and continuous sequence ofcardinals $\langle\kappa_\xi\mid \xi<\mu\rangle$ with $\kappa_0:=\kappa$ andsuch that, for each $\xi<\mu$, $\kappa_{\xi+1}$ is supercompact. Besides,assume that $\lambda$ is a weakly compact cardinal with$\sup_{\xi<\mu}\kappa_\xi<\lambda$. Let $\Theta\geq\lambda$ be a cardinal with$\mathrm{cof}(\Theta)>\kappa$. Assuming the $\mathrm{GCH}_{\geq\kappa}$, weconstruct a generic extension where $\kappa$ is strong limit,$\mathrm{cof}(\kappa)=\mu$, $2^\kappa= \Theta$ and both $\mathrm{TP}(\kappa^+)$and $\mathrm{TP}(\kappa^{++})$ hold. Further, in this model there is a verygood and a bad scale at $\kappa$. This generalizes the main results of [Sin16a]and [FHS18].

The tree property at first and double successors of singular cardinals with an arbitrary gap