combinat-0.2.8.0: Generate and manipulate various combinatorial objects.

Safe Haskell None Haskell2010

Math.Combinat.Trees.Nary

Description

N-ary trees.

Synopsis

# Types

module Data.Tree

data Tree a :: * -> *

Multi-way trees, also known as rose trees.

Constructors

 Node FieldsrootLabel :: alabel valuesubForest :: Forest azero or more child trees

Instances

 Monad Tree Functor Tree Applicative Tree Foldable Tree Traversable Tree Eq a => Eq (Tree a) Data a => Data (Tree a) Read a => Read (Tree a) Show a => Show (Tree a) NFData a => NFData (Tree a) HasNumberOfLeaves (Tree a) HasNumberOfNodes (Tree a) DrawASCII (Tree ()) Typeable (* -> *) Tree

# Regular trees

ternaryTrees :: Int -> [Tree ()] Source

Ternary trees on n nodes (synonym for regularNaryTrees 3)

Arguments

 :: Int degree = number of children of each node -> Int number of nodes -> [Tree ()]

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.

Arguments

 :: [Int] set of allowed number of children -> Int number of nodes -> [Tree ()]

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

autoTabulate RowMajor (Right 5) $map asciiTreeVertical$ map labelNChildrenTree_
$semiRegularTrees [2,3] 2 [ length$ 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

countTernaryTrees :: Integral a => a -> Integer Source

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

countRegularNaryTrees :: (Integral a, Integral b) => a -> b -> Integer Source

We have

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

# "derivation trees"

derivTrees :: [Int] -> [Tree ()] Source

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?

# ASCII drawings

Vertical ASCII drawing of a tree, without labels. Example:

autoTabulate RowMajor (Right 5) $map asciiTreeVertical_$ regularNaryTrees 2 4

Nodes are denoted by @, leaves by *.

asciiTreeVertical :: Show a => Tree a -> ASCII Source

Prints all labels. Example:

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

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

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

# Graphviz drawing

Arguments

 :: Show a => Bool reverse the direction of the arrow -> String name of the graph -> Tree a -> Dot

Generates graphviz .dot file from a tree. The first argument is the name of the graph.

Arguments

 :: Show a => Bool make the individual trees clustered subgraphs -> Bool reverse the direction of the arrows -> String name of the graph -> Forest a -> Dot

Generates graphviz .dot file from a forest. The first argument tells whether to make the individual trees clustered subgraphs; the second is the name of the graph.

# Classifying nodes

classifyTreeNode :: Tree a -> Either a a Source

Left is leaf, Right is node

# Left and right spines

leftSpine :: Tree a -> ([a], a) Source

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

leftSpine_ :: Tree a -> [a] Source

The leftmost spine without the leaf node

rightSpine :: Tree a -> ([a], a) Source

rightSpine_ :: Tree a -> [a] Source

The length (number of edges) on the left spine

leftSpineLength tree == length (leftSpine_ tree)

# Unique labels

addUniqueLabelsTree :: Tree a -> Tree (a, Int) Source

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

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

# Labelling by depth

labelDepthTree :: Tree a -> Tree (a, Int) Source

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

# Labelling by number of children

labelNChildrenTree :: Tree a -> Tree (a, Int) Source

Attaches the number of children to each node.