  The Chern-minimal surfaces in Hermitian surface play a similar role as
minimal surfaces in K\"ahler surface (see \cite{[PX-21]}) from the viewpoint of
submanifolds. This paper studies the compactness of Chern-minimal surfaces. We
prove that any sequence $\{f_n\}$ of conformal Chern-minimal maps from closed
Riemann surface $(\Sigma,\emph{\texttt{j}})$ into a compact Hermitian surface
$(M, J, h)$ with bounded area has a bubble tree limit, which consisting of a
Chern-minimal map $f_0$ from $\Sigma$ into $M$ and a finite set of
Chern-minimal maps from $S^2$ into $M$. We also show that the limit preserves
area and homotopy class.
