test.clj

(fn-1 [{:id 1
  :name "Foo"
  :children [{:id 2
              :name "Bar"
              :children [{:id 3
                          :name "Baz"}
                         {:id 4
                          :name "XXX"}]}
             {:id 3
              :name "Baz"}]}
 {:id 5
  :name "YYY"
  :children [{:id 4
              :name "XXX"}
             {:id 2
              :name "Bar"
              :children [{:id 3
                          :name "Baz"}
                         {:id 4
                          :name "XXX"}]}]}]) =>

[["Foo"]
["Foo" "Bar"]
["Foo" "Bar" "Baz"]
["Foo" "Bar" "XXX"]
["Foo" "Baz"]
["YYY"]
["YYY" "XXX"]
["YYY" "Bar"]
["YYY" "Bar" "Baz"]
["YYY" "Bar" "XXX"]]