Then f must leave some point of M fixed. Hence every map of a tree-like continuum that is homotopic to the identity has a fixed point. This result answers a question of D.P. Bellamy. My argument is somewhat similar to L.E.J. Brouwer's proof that a 2-sphere cannot admit a continuous nonvanishing tangent vector field.
Many related problems remain unsolved.
At the center of this area is the classical problem of extending the Brouwer fixed-point theorem to all nonseparating plane continua.