I will talk about automorphism groups of rooted trees, and, more generally, isometry groups of ultrametric spaces. I will be particularly interested in properties such as property (FA'), property (FA), uncountable strong cofinality and ample generics. First, I will present results relating the last two properties to the behaviour of the algebraic closures of finite sets in the case of countable rooted trees. Then I will focus on the structure of isometry groups of Polish ultrametric spaces, giving an explicit description of ultrametric spaces called W-spaces in terms of wreath products. Building on the this, and the results for trees, I will characterize W-spaces having uncountable strong cofinality.