  In this paper we introduce the notion of pre-simple-minded collection
(pre-SMC) of type $\mathbb{A}$ in the bounded derived categories
$\mathcal{D}^{b} (${\rm{\textbf{T}}}$_p)$ of tube categories $\textbf{T}_{p}$
of rank $p$. This provides an effective approach to classify the hearts in
$\mathcal{D}^{b} (${\rm{\textbf{T}}}$_p)$. We then use this classification to
prove the exchange graph of hearts in $\mathcal{D}^{b} (${\rm{\textbf{T}}}$_p)$
is connnected. Further, we classify the Ext-quivers of hearts in
$\mathcal{D}^{b} (${\rm{\textbf{T}}}$_p)$. As an application, we show that the
space of Bridgeland stability conditions on
$\mathcal{D}^{b}(${\rm{\textbf{T}}}$_p)$ is connected.
