99 questions/61 to 69

This is part of Ninety-Nine Haskell Problems, based on Ninety-Nine Prolog Problems.

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Binary trees

As defined in problem 54A.

An example tree:

tree4 = Branch 1 (Branch 2 Empty (Branch 4 Empty Empty))
(Branch 2 Empty Empty)

Problem 61

Count the leaves of a binary tree

A leaf is a node with no successors. Write a predicate count_leaves/2 to count them.

Example:
% count_leaves(T,N) :- the binary tree T has N leaves

> count_leaves tree4
2

Solution:

count_leaves  Empty                 = 0
count_leaves (Branch a Empty Empty) = 1
count_leaves (Branch a left  right) = count_leaves left + count_leaves right

Problem 61A

Collect the leaves of a binary tree in a list

A leaf is a node with no successors. Write a predicate leaves/2 to collect them in a list.

Example:
% leaves(T,S) :- S is the list of all leaves of the binary tree T

> leaves tree4
[4, 2]

Solution:

leaves :: Tree a -> [a]
leaves  Empty                 = []
leaves (Branch a Empty Empty) = [a]
leaves (Branch a left  right) = leaves left ++ leaves right

Problem 62

Collect the internal nodes of a binary tree in a list An internal node of a binary tree has either one or two non-empty successors. Write a predicate internals/2 to collect them in a list.

Example:
% internals(T,S) :- S is the list of internal nodes of the binary tree T.

Prelude>internals tree4
Prelude>[1,2]

Solution:

internals :: Tree a -> [a]
internals Empty = []
internals (Branch a Empty Empty) = []
internals (Branch a left right) = [a] ++ (internals left) ++ (internals right)

Problem 62B

Collect the nodes at a given level in a list A node of a binary tree is at level N if the path from the root to the node has length N-1. The root node is at level 1. Write a predicate atlevel/3 to collect all nodes at a given level in a list.

Example:
% atlevel(T,L,S) :- S is the list of nodes of the binary tree T at level L

Prelude>atlevel tree4 2
Prelude>[2,2]

Solution:

atlevel :: Tree a -> Int -> [a]
atlevel t level =
atlevel_iter t 1
where
atlevel_iter Empty _ = []
atlevel_iter (Branch a left right) currentLevel =
if currentLevel==level
then [a]
else if currentLevel > level
then []
else (atlevel_iter left (currentLevel+1))
++ (atlevel_iter right (currentLevel+1))

Another possibility is to decompose the problem:

levels :: Tree a -> [[a]]
levels Empty = repeat []
levels (Branch a l r) = [a] : zipWith (++) (levels l) (levels r)

atlevel :: Tree a -> Int -> [a]
atlevel t n = levels t !! (n-1)

Problem 63

<Problem description>

Example:
<example in lisp>

Solution:

<description of implementation>

Problem 64

Given a binary tree as the usual Prolog term t(X,L,R) (or nil). As a preparation for drawing the tree, a layout algorithm is required to determine the position of each node in a rectangular grid. Several layout methods are conceivable, one of them is shown in the illustration below: In this layout strategy, the position of a node v is obtained by the following two rules:

• x(v) is equal to the position of the node v in the inorder sequence
• y(v) is equal to the depth of the node v in the tree

Write a function to annotate each node of the tree with a position, where (1,1) in the top left corner or the rectangle bounding the drawn tree.

Here is the example tree from the above illustration:

tree64 = Branch 'n'
(Branch 'k'
(Branch 'c'
(Branch 'a' Empty Empty)
(Branch 'h'
(Branch 'g'
(Branch 'e' Empty Empty)
Empty
)
Empty
)
)
(Branch 'm' Empty Empty)
)
(Branch 'u'
(Branch 'p'
Empty
(Branch 's'
(Branch 'q' Empty Empty)
Empty
)
)
Empty
)

> layout tree64
Branch ('n',(8,1)) (Branch ('k',(6,2)) (Branch ('c',(2,3)) ...

Solution:

type Pos = (Int, Int)

layout :: Tree a -> Tree (a, Pos)
layout t = fst (layoutAux 1 1 t)
where layoutAux x y Empty = (Empty, x)
layoutAux x y (Branch a l r) = (Branch (a, (x',y)) l' r', x'')
where (l', x')  = layoutAux x (y+1) l
(r', x'') = layoutAux (x'+1) (y+1) r

The auxiliary function is passed the x-coordinate for the left-most node of the subtree, the y-coordinate for the root of the subtree, and the subtree itself. It returns the subtree annotated with positions, plus the count of Branch nodes in the subtree.

Problem 65

An alternative layout method is depicted in the illustration below: Find out the rules and write the corresponding function. Hint: On a given level, the horizontal distance between neighboring nodes is constant.

Use the same conventions as in problem P64 and test your function in an appropriate way.

Here is the example tree from the above illustration:

tree65 = Branch 'n'
(Branch 'k'
(Branch 'c'
(Branch 'a' Empty Empty)
(Branch 'e'
(Branch 'd' Empty Empty)
(Branch 'g' Empty Empty)
)
)
(Branch 'm' Empty Empty)
)
(Branch 'u'
(Branch 'p'
Empty
(Branch 'q' Empty Empty)
)
Empty
)

> layout tree65
Branch ('n',(15,1)) (Branch ('k',(7,2)) (Branch ('c',(3,3)) ...

Solution:

layout :: Tree a -> Tree (a, Pos)
layout t = layoutAux x1 1 sep1 t
where d = depth t
ld = leftdepth t
x1 = 2^(d-1) - 2^(d-ld) + 1
sep1 = 2^(d-2)
layoutAux x y sep Empty = Empty
layoutAux x y sep (Branch a l r) =
Branch (a, (x,y))
(layoutAux (x-sep) (y+1) (sep `div` 2) l)
(layoutAux (x+sep) (y+1) (sep `div` 2) r)

depth :: Tree a -> Int
depth Empty = 0
depth (Branch a l r) = max (depth l) (depth r) + 1

leftdepth :: Tree a -> Int
leftdepth Empty = 0
leftdepth (Branch a l r) = leftdepth l + 1

The auxiliary function is passed the x- and y-coordinates for the root of the subtree, the horizontal separation between the root and its child nodes, and the subtree itself. It returns the subtree annotated with positions.

Problem 66

Yet another layout strategy is shown in the illustration below: The method yields a very compact layout while maintaining a certain symmetry in every node. Find out the rules and write the corresponding Prolog predicate. Hint: Consider the horizontal distance between a node and its successor nodes. How tight can you pack together two subtrees to construct the combined binary tree?

Use the same conventions as in problem P64 and P65 and test your predicate in an appropriate way. Note: This is a difficult problem. Don't give up too early!

Which layout do you like most?

> layout tree65
Branch ('n',(5,1)) (Branch ('k',(3,2)) (Branch ('c',(2,3)) ...

Solution:

layout :: Tree a -> Tree (a, Pos)
layout t = t'
where (l, t', r) = layoutAux x1 1 t
x1 = maximum l + 1

layoutAux :: Int -> Int -> Tree a -> ([Int], Tree (a, Pos), [Int])
layoutAux x y Empty = ([], Empty, [])
layoutAux x y (Branch a l r) = (ll', Branch (a, (x,y)) l' r', rr')
where (ll, l', lr) = layoutAux (x-sep) (y+1) l
(rl, r', rr) = layoutAux (x+sep) (y+1) r
sep = maximum (0:zipWith (+) lr rl) `div` 2 + 1
ll' = 0 : overlay (map (+sep) ll) (map (subtract sep) rl)
rr' = 0 : overlay (map (+sep) rr) (map (subtract sep) lr)

-- overlay xs ys = xs padded out to at least the length of ys
-- using any extra elements of ys
overlay :: [a] -> [a] -> [a]
overlay [] ys = ys
overlay xs [] = xs
overlay (x:xs) (y:ys) = x : overlay xs ys

The auxiliary function is passed the x- and y-coordinates for the root of the subtree and the subtree itself. It returns

• a list of distances the laid-out tree extends to the left at each level,
• the subtree annotated with positions, and
• a list of distances the laid-out tree extends to the right at each level.

These distances are usually positive, but may be 0 or negative in the case of a skewed tree. To put two subtrees side by side, we must determine the least even separation so that they do not overlap on any level. Having determined the separation, we can compute the extents of the composite tree.

The definitions of layout and its auxiliary function use local recursion to compute the x-coordinates. This works because nothing else depends on these coordinates.

Problem 67

<Problem description>

Example:
<example in lisp>

Solution:

<description of implementation>

Problem 68

<Problem description>

Example:
<example in lisp>

Solution:

<description of implementation>

Problem 69

<Problem description>

Example:
<example in lisp>