  Homeomorphism groups of generalized Wa\.zewski dendrites act on the infinite
countable set of branch points of the dendrite and thus have a nice Polish
topology. In this paper, we study them in the light of this Polish topology.
The group of the universal Wa\.zewski dendrite $D_\infty$ is more
characteristic than the others because it is the unique one with a dense
conjugacy class. For this group $G_\infty$, we show some of its topological
properties like existence of a comeager conjugacy class, the Steinhaus
property, automatic continuity and the small index subgroup property. Moreover,
we identify the universal minimal flow of $G_\infty$. This allows us to prove
that point-stabilizers in $G_\infty$ are amenable and to describe the universal
Furstenberg boundary of $G_\infty$.
