How to treat input that inherently has a tree structure? When having a single vector you use an MLP neural network
When having a 2D structure you use a CNN neural network
When having a sequence you use a RNN neural network

Now you have preprocessed an instance and the result is a tree structure.
Let's say for simplicity that the tree structure is always the same tree; only the node values differ among instances.

What kind of neural network architecture would be required to consume the information of a tree structure but also leveraging the connections between the tree nodes?
As @Emre mentioned, RNN is a good option. It's worth noting that if the number of possible nodes in each tree is the same or at least has the same upper bound, you could use literally any architecture you want and just pass in the adjacency matrix. Alternatively, you could build an intermediate model to convert your graph into a graph embedding and then once again, you can do basically whatever you want with that.

A pretty big piece of potentially important information here is what you are trying to accomplish, which could have significant consequences for how you want to represent your inputs.