Tree-like continua and exactly $k$-to-$1$ functions

To answer a question of Nadler and Ward, $k$-to-$1$ maps from tree-like continua onto tree-like continua are constructed, for $k > 2$. It is shown that certain arc-like continua cannot be the domain of any $2$-to-$1$ map and that certain tree-like continua cannot be the image of any $2$-to-$1$ map (defined on continua) but it is unknown if any indecomposable arc-like continuum can be the domain or any tree-like continuum the image of a $2$-to-$1$ map.