Making three membered ring using different reactants
I think that both reactions involve carbene as common intermediate which then is the species involved in the subsequent cyclopropanation reaction.
These are the two equations which lead to the formation of the carbene:
$$\ce{KOH + CHCl3 -&gt;  :CCl2 + KCl + H2O}$$
$$\ce{Cl3CCO2Na -&gt;  :CCl2 + NaCl + CO2}$$
Subsequently, the alkene undergoes cyclopropanation by dichlorocarbene.
In the second case, heat is required in order to promote decarboxylation, which is entropically favored.