Data.Graph

Finite Graphs

The Graph type is an adjacency list representation of a finite, directed graph with vertices of type Int.

The SCC type represents a strongly-connected component of a graph.

Implementation

The implementation is based on

• Structuring Depth-First Search Algorithms in Haskell, by David King and John Launchbury, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.6526

Graphs

type Graph = Array Vertex [Vertex]

Adjacency list representation of a graph, mapping each vertex to its list of successors.

type Bounds = (Vertex, Vertex)

The bounds of an Array.

type Edge = (Vertex, Vertex)

An edge from the first vertex to the second.

type Vertex = Int

Abstract representation of vertices.

type Table a = Array Vertex a

Table indexed by a contiguous set of vertices.

Note: This is included for backwards compatibility.

Graph Construction

graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex)

\(O((V+E) \log V)\). Build a graph from a list of nodes uniquely identified by keys, with a list of keys of nodes this node should have edges to.

This function takes an adjacency list representing a graph with vertices of type key labeled by values of type node and produces a Graph-based representation of that list. The Graph result represents the shape of the graph, and the functions describe a) how to retrieve the label and adjacent vertices of a given vertex, and b) how to retrieve a vertex given a key.

(graph, nodeFromVertex, vertexFromKey) = graphFromEdges edgeList

• graph :: Graph is the raw, array based adjacency list for the graph.
• nodeFromVertex :: Vertex -> (node, key, [key]) returns the node associated with the given 0-based Int vertex; see warning below. This runs in \(O(1)\) time.
• vertexFromKey :: key -> Maybe Vertex returns the Int vertex for the key if it exists in the graph, Nothing otherwise. This runs in \(O(\log V)\) time.

To safely use this API you must either extract the list of vertices directly from the graph or first call vertexFromKey k to check if a vertex corresponds to the key k. Once it is known that a vertex exists you can use nodeFromVertex to access the labelled node and adjacent vertices. See below for examples.

Note: The out-list may contain keys that don't correspond to nodes of the graph; they are ignored.

Warning: The nodeFromVertex function will cause a runtime exception if the given Vertex does not exist.

Examples

An empty graph.

(graph, nodeFromVertex, vertexFromKey) = graphFromEdges []
graph = array (0,-1) []

A graph where the out-list references unspecified nodes ('c'), these are ignored.

(graph, _, _) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c'])]
array (0,1) [(0,[1]),(1,[])]

A graph with 3 vertices: ("a") -> ("b") -> ("c")

(graph, nodeFromVertex, vertexFromKey) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c']), ("c", 'c', [])]
graph == array (0,2) [(0,[1]),(1,[2]),(2,[])]
nodeFromVertex 0 == ("a",'a',"b")
vertexFromKey 'a' == Just 0

Get the label for a given key.

let getNodePart (n, _, _) = n
(graph, nodeFromVertex, vertexFromKey) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c']), ("c", 'c', [])]
getNodePart . nodeFromVertex <$> vertexFromKey 'a' == Just "A"

graphFromEdges' :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]))

\(O((V+E) \log V)\). Identical to graphFromEdges, except that the return value does not include the function which maps keys to vertices. This version of graphFromEdges is for backwards compatibility.

buildG :: Bounds -> [Edge] -> Graph

\(O(V+E)\). Build a graph from a list of edges.

Warning: This function will cause a runtime exception if a vertex in the edge list is not within the given Bounds.

Examples

buildG (0,-1) [] == array (0,-1) []
buildG (0,2) [(0,1), (1,2)] == array (0,1) [(0,[1]),(1,[2])]
buildG (0,2) [(0,1), (0,2), (1,2)] == array (0,2) [(0,[2,1]),(1,[2]),(2,[])]

Graph Properties

vertices :: Graph -> [Vertex]

\(O(V)\). Returns the list of vertices in the graph.

Examples

vertices (buildG (0,-1) []) == []

vertices (buildG (0,2) [(0,1),(1,2)]) == [0,1,2]

edges :: Graph -> [Edge]

\(O(V+E)\). Returns the list of edges in the graph.

Examples

edges (buildG (0,-1) []) == []

edges (buildG (0,2) [(0,1),(1,2)]) == [(0,1),(1,2)]

outdegree :: Graph -> Array Vertex Int

\(O(V+E)\). A table of the count of edges from each node.

Examples

outdegree (buildG (0,-1) []) == array (0,-1) []

outdegree (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,1),(1,1),(2,0)]

indegree :: Graph -> Array Vertex Int

\(O(V+E)\). A table of the count of edges into each node.

Examples

indegree (buildG (0,-1) []) == array (0,-1) []

indegree (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,0),(1,1),(2,1)]

Graph Transformations

transposeG :: Graph -> Graph

\(O(V+E)\). The graph obtained by reversing all edges.

Examples

transposeG (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,[]),(1,[0]),(2,[1])]

Graph Algorithms

dfs :: Graph -> [Vertex] -> Forest Vertex

\(O(V+E)\). A spanning forest of the part of the graph reachable from the listed vertices, obtained from a depth-first search of the graph starting at each of the listed vertices in order.

dff :: Graph -> [Tree Vertex]

\(O(V+E)\). A spanning forest of the graph, obtained from a depth-first search of the graph starting from each vertex in an unspecified order.

topSort :: Graph -> [Vertex]

\(O(V+E)\). A topological sort of the graph. The order is partially specified by the condition that a vertex i precedes j whenever j is reachable from i but not vice versa.

Note: A topological sort exists only when there are no cycles in the graph. If the graph has cycles, the output of this function will not be a topological sort. In such a case consider using scc.

reverseTopSort :: Graph -> [Vertex]

\(O(V+E)\). Reverse ordering of topSort.

See note in topSort.

Since: 0.6.4

components :: Graph -> [Tree Vertex]

\(O(V+E)\). The connected components of a graph. Two vertices are connected if there is a path between them, traversing edges in either direction.

scc :: Graph -> [Tree Vertex]

\(O(V+E)\). The strongly connected components of a graph, in reverse topological order.

Examples

scc (buildG (0,3) [(3,1),(1,2),(2,0),(0,1)])
  == [Node {rootLabel = 0, subForest = [Node {rootLabel = 1, subForest = [Node {rootLabel = 2, subForest = []}]}]}
     ,Node {rootLabel = 3, subForest = []}]

bcc :: Graph -> [Tree [Vertex]]

\(O(V+E)\). The biconnected components of a graph. An undirected graph is biconnected if the deletion of any vertex leaves it connected.

The input graph is expected to be undirected, i.e. for every edge in the graph the reverse edge is also in the graph. If the graph is not undirected the output is arbitrary.

reachable :: Graph -> Vertex -> [Vertex]

\(O(V+E)\). Returns the list of vertices reachable from a given vertex.

Examples

reachable (buildG (0,0) []) 0 == [0]

reachable (buildG (0,2) [(0,1), (1,2)]) 0 == [0,1,2]

path :: Graph -> Vertex -> Vertex -> Bool

\(O(V+E)\). Returns True if the second vertex reachable from the first.

Examples

path (buildG (0,0) []) 0 0 == True

path (buildG (0,2) [(0,1), (1,2)]) 0 2 == True

path (buildG (0,2) [(0,1), (1,2)]) 2 0 == False

Strongly Connected Components

data SCC vertex

Strongly connected component.

Constructors

AcyclicSCC vertex

A single vertex that is not in any cycle.

CyclicSCC [vertex]

A maximal set of mutually reachable vertices.

Instances

Foldable SCC

Since: 0.5.9

Defined in Data.Graph

Methods

fold :: Monoid m => SCC m -> m

foldMap :: Monoid m => (a -> m) -> SCC a -> m

foldMap' :: Monoid m => (a -> m) -> SCC a -> m

foldr :: (a -> b -> b) -> b -> SCC a -> b

foldr' :: (a -> b -> b) -> b -> SCC a -> b

foldl :: (b -> a -> b) -> b -> SCC a -> b

foldl' :: (b -> a -> b) -> b -> SCC a -> b

foldr1 :: (a -> a -> a) -> SCC a -> a

foldl1 :: (a -> a -> a) -> SCC a -> a

toList :: SCC a -> [a]

null :: SCC a -> Bool

length :: SCC a -> Int

elem :: Eq a => a -> SCC a -> Bool

maximum :: Ord a => SCC a -> a

minimum :: Ord a => SCC a -> a

sum :: Num a => SCC a -> a

product :: Num a => SCC a -> a

Eq1 SCC

Since: 0.5.9

Defined in Data.Graph

Methods

liftEq :: (a -> b -> Bool) -> SCC a -> SCC b -> Bool

Read1 SCC

Since: 0.5.9

Defined in Data.Graph

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (SCC a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [SCC a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (SCC a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [SCC a]

Show1 SCC

Since: 0.5.9

Defined in Data.Graph

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> SCC a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [SCC a] -> ShowS

Traversable SCC

Since: 0.5.9

Defined in Data.Graph

Methods

traverse :: Applicative f => (a -> f b) -> SCC a -> f (SCC b)

sequenceA :: Applicative f => SCC (f a) -> f (SCC a)

mapM :: Monad m => (a -> m b) -> SCC a -> m (SCC b)

sequence :: Monad m => SCC (m a) -> m (SCC a)

Functor SCC

Since: 0.5.4

Defined in Data.Graph

Methods

fmap :: (a -> b) -> SCC a -> SCC b

(<$) :: a -> SCC b -> SCC a

Generic1 SCC

Defined in Data.Graph

Associated Types

type Rep1 SCC :: k -> Type

Methods

from1 :: forall (a :: k). SCC a -> Rep1 SCC a

to1 :: forall (a :: k). Rep1 SCC a -> SCC a

Lift vertex => Lift (SCC vertex :: Type)

Since: 0.6.6

Defined in Data.Graph

Methods

lift :: Quote m => SCC vertex -> m Exp

liftTyped :: forall (m :: Type -> Type). Quote m => SCC vertex -> Code m (SCC vertex)

Data vertex => Data (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> SCC vertex -> c (SCC vertex)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (SCC vertex)

toConstr :: SCC vertex -> Constr

dataTypeOf :: SCC vertex -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (SCC vertex))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (SCC vertex))

gmapT :: (forall b. Data b => b -> b) -> SCC vertex -> SCC vertex

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> SCC vertex -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> SCC vertex -> r

gmapQ :: (forall d. Data d => d -> u) -> SCC vertex -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> SCC vertex -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex)

Generic (SCC vertex)

Defined in Data.Graph

Associated Types

type Rep (SCC vertex) :: Type -> Type

Methods

from :: SCC vertex -> Rep (SCC vertex) x

to :: Rep (SCC vertex) x -> SCC vertex

Read vertex => Read (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

Methods

readsPrec :: Int -> ReadS (SCC vertex)

readList :: ReadS [SCC vertex]

readPrec :: ReadPrec (SCC vertex)

readListPrec :: ReadPrec [SCC vertex]

Show vertex => Show (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

Methods

showsPrec :: Int -> SCC vertex -> ShowS

show :: SCC vertex -> String

showList :: [SCC vertex] -> ShowS

NFData a => NFData (SCC a)

Defined in Data.Graph

Methods

rnf :: SCC a -> ()

Eq vertex => Eq (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

Methods

(==) :: SCC vertex -> SCC vertex -> Bool

(/=) :: SCC vertex -> SCC vertex -> Bool

type Rep1 SCC

Since: 0.5.9

Defined in Data.Graph

type Rep1 SCC = D1 ('MetaData "SCC" "Data.Graph" "containers-0.6.7-inplace" 'False) (C1 ('MetaCons "AcyclicSCC" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1) :+: C1 ('MetaCons "CyclicSCC" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec1 List)))

type Rep (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

type Rep (SCC vertex) = D1 ('MetaData "SCC" "Data.Graph" "containers-0.6.7-inplace" 'False) (C1 ('MetaCons "AcyclicSCC" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 vertex)) :+: C1 ('MetaCons "CyclicSCC" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 [vertex])))

Construction

stronglyConnComp

Arguments

:: Ord key

=> [(node, key, [key])]

The graph: a list of nodes uniquely identified by keys, with a list of keys of nodes this node has edges to. The out-list may contain keys that don't correspond to nodes of the graph; such edges are ignored.

-> [SCC node]

\(O((V+E) \log V)\). The strongly connected components of a directed graph, reverse topologically sorted.

Examples

stronglyConnComp [("a",0,[1]),("b",1,[2,3]),("c",2,[1]),("d",3,[3])]
  == [CyclicSCC ["d"],CyclicSCC ["b","c"],AcyclicSCC "a"]

stronglyConnCompR

Arguments

:: Ord key

=> [(node, key, [key])]

The graph: a list of nodes uniquely identified by keys, with a list of keys of nodes this node has edges to. The out-list may contain keys that don't correspond to nodes of the graph; such edges are ignored.

-> [SCC (node, key, [key])]

Reverse topologically sorted

\(O((V+E) \log V)\). The strongly connected components of a directed graph, reverse topologically sorted. The function is the same as stronglyConnComp, except that all the information about each node retained. This interface is used when you expect to apply SCC to (some of) the result of SCC, so you don't want to lose the dependency information.

Examples

stronglyConnCompR [("a",0,[1]),("b",1,[2,3]),("c",2,[1]),("d",3,[3])]
 == [CyclicSCC [("d",3,[3])],CyclicSCC [("b",1,[2,3]),("c",2,[1])],AcyclicSCC ("a",0,[1])]

Conversion

flattenSCC :: SCC vertex -> [vertex]

The vertices of a strongly connected component.

flattenSCCs :: [SCC a] -> [a]

The vertices of a list of strongly connected components.

Trees

data Tree a

Non-empty, possibly infinite, multi-way trees; also known as rose trees.

Constructors

Node a [Tree a]

Instances

MonadFix Tree

Since: 0.5.11

Defined in Data.Tree

Methods

mfix :: (a -> Tree a) -> Tree a

MonadZip Tree

Since: 0.5.10.1

Defined in Data.Tree

Methods

mzip :: Tree a -> Tree b -> Tree (a, b)

mzipWith :: (a -> b -> c) -> Tree a -> Tree b -> Tree c

munzip :: Tree (a, b) -> (Tree a, Tree b)

Foldable Tree

Folds in preorder

Defined in Data.Tree

Methods

fold :: Monoid m => Tree m -> m

foldMap :: Monoid m => (a -> m) -> Tree a -> m

foldMap' :: Monoid m => (a -> m) -> Tree a -> m

foldr :: (a -> b -> b) -> b -> Tree a -> b

foldr' :: (a -> b -> b) -> b -> Tree a -> b

foldl :: (b -> a -> b) -> b -> Tree a -> b

foldl' :: (b -> a -> b) -> b -> Tree a -> b

foldr1 :: (a -> a -> a) -> Tree a -> a

foldl1 :: (a -> a -> a) -> Tree a -> a

toList :: Tree a -> [a]

null :: Tree a -> Bool

length :: Tree a -> Int

elem :: Eq a => a -> Tree a -> Bool

maximum :: Ord a => Tree a -> a

minimum :: Ord a => Tree a -> a

sum :: Num a => Tree a -> a

product :: Num a => Tree a -> a

Foldable1 Tree

Folds in preorder

Since: 0.6.7

Defined in Data.Tree

Methods

fold1 :: Semigroup m => Tree m -> m

foldMap1 :: Semigroup m => (a -> m) -> Tree a -> m

foldMap1' :: Semigroup m => (a -> m) -> Tree a -> m

toNonEmpty :: Tree a -> NonEmpty a

maximum :: Ord a => Tree a -> a

minimum :: Ord a => Tree a -> a

head :: Tree a -> a

last :: Tree a -> a

foldrMap1 :: (a -> b) -> (a -> b -> b) -> Tree a -> b

foldlMap1' :: (a -> b) -> (b -> a -> b) -> Tree a -> b

foldlMap1 :: (a -> b) -> (b -> a -> b) -> Tree a -> b

foldrMap1' :: (a -> b) -> (a -> b -> b) -> Tree a -> b

Eq1 Tree

Since: 0.5.9

Defined in Data.Tree

Methods

liftEq :: (a -> b -> Bool) -> Tree a -> Tree b -> Bool

Ord1 Tree

Since: 0.5.9

Defined in Data.Tree

Methods

liftCompare :: (a -> b -> Ordering) -> Tree a -> Tree b -> Ordering

Read1 Tree

Since: 0.5.9

Defined in Data.Tree

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Tree a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Tree a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Tree a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Tree a]

Show1 Tree

Since: 0.5.9

Defined in Data.Tree

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Tree a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Tree a] -> ShowS

Traversable Tree

Defined in Data.Tree

Methods

traverse :: Applicative f => (a -> f b) -> Tree a -> f (Tree b)

sequenceA :: Applicative f => Tree (f a) -> f (Tree a)

mapM :: Monad m => (a -> m b) -> Tree a -> m (Tree b)

sequence :: Monad m => Tree (m a) -> m (Tree a)

Applicative Tree

Defined in Data.Tree

Methods

pure :: a -> Tree a

(<*>) :: Tree (a -> b) -> Tree a -> Tree b

liftA2 :: (a -> b -> c) -> Tree a -> Tree b -> Tree c

(*>) :: Tree a -> Tree b -> Tree b

(<*) :: Tree a -> Tree b -> Tree a

Functor Tree

Defined in Data.Tree

Methods

fmap :: (a -> b) -> Tree a -> Tree b

(<$) :: a -> Tree b -> Tree a

Monad Tree

Defined in Data.Tree

Methods

(>>=) :: Tree a -> (a -> Tree b) -> Tree b

(>>) :: Tree a -> Tree b -> Tree b

return :: a -> Tree a

Generic1 Tree

Defined in Data.Tree

Associated Types

type Rep1 Tree :: k -> Type

Methods

from1 :: forall (a :: k). Tree a -> Rep1 Tree a

to1 :: forall (a :: k). Rep1 Tree a -> Tree a

Lift a => Lift (Tree a :: Type)

Since: 0.6.6

Defined in Data.Tree

Methods

lift :: Quote m => Tree a -> m Exp

liftTyped :: forall (m :: Type -> Type). Quote m => Tree a -> Code m (Tree a)

Data a => Data (Tree a)

Defined in Data.Tree

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Tree a -> c (Tree a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Tree a)

toConstr :: Tree a -> Constr

dataTypeOf :: Tree a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Tree a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Tree a))

gmapT :: (forall b. Data b => b -> b) -> Tree a -> Tree a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Tree a -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Tree a -> r

gmapQ :: (forall d. Data d => d -> u) -> Tree a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Tree a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a)

Generic (Tree a)

Defined in Data.Tree

Associated Types

type Rep (Tree a) :: Type -> Type

Methods

from :: Tree a -> Rep (Tree a) x

to :: Rep (Tree a) x -> Tree a

Read a => Read (Tree a)

Defined in Data.Tree

Methods

readsPrec :: Int -> ReadS (Tree a)

readList :: ReadS [Tree a]

readPrec :: ReadPrec (Tree a)

readListPrec :: ReadPrec [Tree a]

Show a => Show (Tree a)

Defined in Data.Tree

Methods

showsPrec :: Int -> Tree a -> ShowS

show :: Tree a -> String

showList :: [Tree a] -> ShowS

NFData a => NFData (Tree a)

Defined in Data.Tree

Methods

rnf :: Tree a -> ()

Eq a => Eq (Tree a)

Defined in Data.Tree

Methods

(==) :: Tree a -> Tree a -> Bool

(/=) :: Tree a -> Tree a -> Bool

Ord a => Ord (Tree a)

Since: 0.6.5

Defined in Data.Tree

Methods

compare :: Tree a -> Tree a -> Ordering

(<) :: Tree a -> Tree a -> Bool

(<=) :: Tree a -> Tree a -> Bool

(>) :: Tree a -> Tree a -> Bool

(>=) :: Tree a -> Tree a -> Bool

max :: Tree a -> Tree a -> Tree a

min :: Tree a -> Tree a -> Tree a

type Rep1 Tree

Since: 0.5.8

Defined in Data.Tree

type Rep1 Tree = D1 ('MetaData "Tree" "Data.Tree" "containers-0.6.7-inplace" 'False) (C1 ('MetaCons "Node" 'PrefixI 'True) (S1 ('MetaSel ('Just "rootLabel") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1 :*: S1 ('MetaSel ('Just "subForest") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (List :.: Rec1 Tree)))

type Rep (Tree a)

Since: 0.5.8

Defined in Data.Tree

type Rep (Tree a) = D1 ('MetaData "Tree" "Data.Tree" "containers-0.6.7-inplace" 'False) (C1 ('MetaCons "Node" 'PrefixI 'True) (S1 ('MetaSel ('Just "rootLabel") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Just "subForest") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 [Tree a])))

type Forest a = [Tree a]

This type synonym exists primarily for historical reasons.