Non-additivity of the fixed point property for tree-like continua
We investigate the fixed point property for tree-like continua that are unions of tree-like continua. We obtain a positive result if finitely many tree-like continua with the fixed point property have dendrites for pairwise intersections. Using Bellamy's seminal example, we define (i) a countable wedge $\hat X$ of tree-like continua, each having the fpp, and $\hat X$ admitting a fixed-point-free homeomorphism, and (ii) two tree-like continua $H$ and $K$ such that $H$, $K$, and $H\cap K$ have the fixed point property, but $H\cup K$ admits a fixed-point-free homeomorphism. In an appendix we verify some of the properties of Bellamy's continuum.