Ternary trees on n nodes (synonym for regularNaryTrees 3)

regularNaryTrees d n returns the list of (rooted) trees on n nodes where each node has exactly d children. Note that the leaves do not count in n. Naive algorithm.

All trees on n nodes where the number of children of all nodes is in element of the given set. Example:

 mapM_ printTreeVertical 
  $ map labelNChildrenTree_ 
  $ semiRegularTrees [2,3] n

 [ length $ semiRegularTrees [2,3] n | n<-[0..] ] == [1,2,10,66,498,4066,34970,312066,2862562,26824386,...]

The latter sequence is A027307 in OEIS: https://oeis.org/A027307

Remark: clearly, we have

 semiRegularTrees [d] n == regularNaryTrees d n

# = \frac {1} {(2n+1} \binom {3n} {n}

We have

 length (regularNaryTrees d n) == countRegularNaryTrees d n == \frac {1} {(d-1)n+1} \binom {dn} {n} 

Computes the set of equivalence classes of rooted trees (in the sense that the leaves of a node are unordered) with n = length ks leaves where the set of heights of the leaves matches the given set of numbers. The height is defined as the number of edges from the leaf to the root.

TODO: better name?

Vertical ASCII drawing of a tree, without labels.

Example:

 mapM_ printTreeVertical_ $ regularNaryTrees 2 3 

Prints all labels.

Example:

 printTreeVertical $ addUniqueLabelsTree_ $ (regularNaryTrees 3 9) !! 666

Prints the labels for the leaves, but not for the nonempty nodes

Nodes are denoted by @, leaves by *.

Nodes are denoted by (label), leaves by label.

Nodes are denoted by @, leaves by label.

Left is leaf, Right is node

The leftmost spine (the second element of the pair is the leaf node)

The leftmost spine without the leaf node

The length (number of edges) on the left spine

 leftSpineLength tree == length (leftSpine_ tree)

Adds unique labels to the nodes (including leaves) of a Tree.

Adds unique labels to the nodes (including leaves) of a Forest

Attaches the depth to each node. The depth of the root is 0.

Attaches the number of children to each node.