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.