semigroups acting as continuous functions on regular rooted trees

Let $T$ be a regular rooted tree. Make $T$ into a metric space by making each edge isometric to the unit interval. What is known about what semigroups can act as continuous functions on $T$ such that each element of the semigroup fixes the root of $T$ and each element maps vertices to vertices? Are there any algebraic restrictions on such semigroups, for example? Can every semigroup arise in this way? Thanks!

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