We introduce a novel approach to tree-to-tree learning, the neural tree transducer (NTT), a top-down depth first context-sensitive tree decoder, which is paired with recursive neural encoders. Our method works purely on tree-to-tree manipulations rather than sequence-to-tree or tree-to-sequence and is able to encode and decode multiple depth trees. We compare our method to sequence-to-sequence models applied to serializations of the trees and show that our method outperforms previous methods for tree-to-tree transduction.
The paper introduces a neural tree decoder architecture for binary trees that conditions the next node prediction on 
representations of its ascendants (encoded with an LSTM recurrent net) and left sibling subtree (encoded with a binary LSTM recursive net) for right sibling nodes.     
To perform tree to tree transduction the input tree is encoded as a vector with a Tree LSTM; correspondences between input and output subtrees are not modelled directly (using e.g. attention) as is done in traditional tree transducers. 
While the term context-sensitive should be used with caution, I do accept the claim here, although the notation used does not make the exposition clear. 
Experimental results show that the architecture performs better at synthetic tree transduction tasks (relabeling, reordering, deletion) than sequence-to-sequence baselines. 
While neural approches to tree-to-tree transduction is an understudied problem, the contributions of this paper are very narrow and it is not shown that the proposed approach will generalize to more expressive models or real-world applications of tree-to-tree transduction. 
Existing neural tree decoders, such as Dong and Lapata or Alvarex-Melis and Jaakkola, could be combined with tree LSTM encoders without any technical innovations and could possibly do as well as the proposed model for the transduction tasks tested - no experiments are performed with existing tree-based decoder architectures. 
Specific comments per section:
1. Unclear what is meant be "equivalent" in first paragraph. 
2. The model does not assign an explicit probability to the tree structure - rather it seems to rely on the distinction between terminal and non-terimal symbols and the restriction to binary trees to know when closing brackets are implied - this is not made clear, and a general model should not have this restriction, as there are many cases where we want to generate non-binary trees.
The production rule notation used is incorrect and confusing, mixing sets with non-terminals and terminal symbols: 
A better notation for the rules in 2.1.1 would be something like S -> P | v | \epsilon; P -> Q R | Q u | u Q | u w, where P, Q, R \in O and u, w \in v.
2.1.2. Splitting production rules as ->_left, ->_right is not standard notation. Rather introduce intermediate non-terminals in the grammar:
O -> O_L O_R; O_L -> a | Q, O_R -> b | Q. 
2.1.3 The context-sensitively here arise when conditioning on the entire left sibling subtree (not just the top non-terimal).
The rules should have a format such as O -> O_L O_R; O_L -> a | Q; \alpha O_R -> \alpha a | \alpha Q, where \alpha is an entire subtree rooted at O_L.
2.1.4 Should be g(x|.) = exp( ), the softmax function includes the normalization which is done in the equation below. 
3. Note that is is possible to restrict the decoder to produce tree structures while keeping a sequential neural architecture. For some tasks sequential decoders do actually produce mostly well-formed trees, given enough training data. 
RNNG encodes completed subtrees recursively, and the stack LSTM encodes the entire partially-produced tree, so it does produce and condition on trees not just sequences. The model in this paper is not more expressive than RNNG, it just encodes somewhat different structural biases, which might or might not be suited for real tasks.  
4. In the examples given, the same set of symbols are used as both terminals and non-terminals. How is the tree structure then predicted by the decoder?
Details about the training setup are missing: How is the training data generated, what is the size of the trees during training (compared to testing)?
4.2 The steep drop in performance between depth 5 and 6 indicates that model is very sensitive to its memorization capacity and might not be generalizing over the given training data.
For real tree-to-tree applications involving these operations, there is good reason to believe that some kind of attention mechanism will be needed over the input tree during decoding. 
Reference should generally be to published proceedings rather than to arxiv where available - e.g. Aharoni and Goldberg, Dong and Lapata, Erguchi et al, Rush et al. For Graehl and Knight there is a published journal paper in Computational Linguistics.