  We give two characterizations of $\mathcal P$-like continua $X$ that do not
have the fixed point property. Both characterizations are stated in terms of
sequences of open covers of $X$ that follow fixed-point-free patterns. We use
these to characterize planar tree-like continua that do not have the fixed
point property in terms of infinite sequences of tree-chains in the plane that
follow fixed-point-free patterns. We also establish a useful relationship
between these tree-chains and commutative simplicial diagrams that we use later
to construct a finite sequence (of any given length) of tree-chains in the
plane that follows a fixed-point-free pattern.
  An earlier characterization of $\mathcal P$-like continua with the fixed
point property was given in 1994 by Feuerbacher based on a 1963 result by
Mioduszewski. The Mioduszewski-Feuerbacher characterization is expressed in
terms of almost commutative inverse diagrams. In contrast, our approach is more
geometric, and it may potentially lead to new methods in the elusive search for
a planar tree-like continuum without the fixed-point property.

Dynamical Systems