In the lecture on counting unlabeled trees, we introduced the planar codes of particular planar embeddings (up to deforming the plane) of rooted trees. (The planar embeddings can also be thought of as fixing the ordering ("from left to right") of the children of all nodes.) The resulting planar code are binary words of length 2, where is the number of edges. In the lecture, we observed that not all binary words can be planar codes of a planar embedding of rooted tree; for example, we have to have the same number of 0's and 1's. In fact, the necessary and sufficient conditions for a binary word of length 2 to be the planar code of some planar embedding of some rooted tree are the following: (1) The number of 0's and 1's is the same. (2) The number of 1's is at least the number of 0's appearing to the left of any position in the sequence. Such binary words are known as Dyck words.

For example, the tree with edge set {{,},{,},{,}}has 2 distinct planar embeddings with planar codes (Dyck words) 101100and 110010. There are also 2 other Dyck words of length 6; think about what trees do they correspond to.

Consider the minimal non-linear tree (see previous problem for a definition) with 00 as the root. Draw some embeddings of in order to convince yourself that they are identical up to relabeling. Therefore, the Dyck word of is uniquely determined. What is the Dyck word of ? Your answer as a comma separated list: How many Dyck words of length 8 are there? Your answer: How many planar embeddings (up to deforming the plane) of rooted trees with 4 edges are there? Your answer: