Universal minimal flows of generalized Wa\.zewski dendrites

We study universal minimal flows of the homeomorphism groups of generalized Wa\.zewski dendrites $W_P$, $P\subset\{3,4,\ldots,\omega\}$. If $P$ is finite, we prove that the universal minimal flow of of the homeomorphism group $H(W_P)$ is metrizable and we compute it explicitly. This answers a question of B. Duchesne. If $P$ is infinite, we show that the universal minimal flow of $H(W_P)$ is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a non-metrizable universal minimal flow with a comeager orbit.