Name for tree where each node has the aggregate of its children
Such tree is called Sum Tree or SumTree, where the property is maintained in bottom up fashion. In general if the value is additional value (node property) for some existing one like AVL it is Sum augmented AVL Tree.

If the property is inherent, like maintaining the prefix sum, it would be Fenwick Tree.

As far as I know there is no established name, but for other functions it is partial statistics tree. preferable name would be Tree augmented with $f$, to make it explicit.