  The spatial fluctuations in the tomographic maps of the redshifted 21-cm
signal from the Cosmic Dawn (CD) crucially depend on the size and distribution
of the regions with gas temperatures larger than the radio background
temperature. In this article, we study the morphological characteristics of
such emission regions and their absorption counterparts using the shape
diagnostic tool SURFGEN2. Using simulated CD brightness temperature cubes of
the 21-cm signal, we find that the emission regions percolate at stages with
the filling factor of the emission regions $FF_{\rm emi}\gtrsim 0.15$.
Percolation of the absorption regions occurs for $FF_{\rm abs}\gtrsim 0.05$.
The largest emission and absorption regions are topologically complex and
highly filamentary for most parts of the CD. The number density of these
regions as a function of the volume shows the power-law nature with the
power-law indexes $\approx -2$ and $-1.6$ for the emission and absorption
regions, respectively. Overall, the planarity, filamentarity and genus increase
with the increase of the volume of both emission and absorption regions.
