We will discuss a class of very interesting one-dimensional metric continua called Wazewski dendrites. The homeomorphism group of a Wazewski dendrite is isomorphic to an automorphism group of a certain countable structure. Using the Kechris-Pestov-Todorcevic correspondence relating topological dynamics and Ramsey theory, we will compute universal minimal flows of those homeomorphism groups of Wazewski dendrites whose universal minimal flow is metrizable.