# Trees as nested tuples
t = ((1, 2), (3, 4), 5)

def count_leaves(tree):
    """Return the number of leaves in a tree.

    >>> count_leaves(t)
    5
    """
    if type(tree) != tuple:
        return 1
    return sum(map(count_leaves, tree))

def map_tree(tree, fn):
    """Return a tree with fn mapped to the leaves of tree.

    >>> map_tree(t, lambda x: x*x)
    ((1, 4), (9, 16), 25)
    """
    if type(tree) != tuple:
        return fn(tree)
    return tuple(map_tree(branch, fn) for branch in tree)

# Tree class (binary trees with internal values)
class Tree(object):
    """A tree with internal values."""

    def __init__(self, entry, left=None, right=None):
        self.entry = entry
        self.left = left
        self.right = right

    def __repr__(self):
        args = repr(self.entry)
        if self.left or self.right:
            args += ', {0}, {1}'.format(repr(self.left),
                                        repr(self.right))
        return 'Tree({0})'.format(args)

def fib_tree(n):
    """Return a Tree that represents a recursive Fibonacci
    calculation.

    >>> fib_tree(3)
    Tree(1, Tree(0), Tree(1))
    """
    if n == 1:
        return Tree(0)
    if n == 2:
        return Tree(1)
    left = fib_tree(n - 2)
    right = fib_tree(n - 1)
    return Tree(left.entry + right.entry, left, right)

def count_entries(tree):
    """Return the number of entries in a Tree.

    >>> count_entries(fib_tree(6))
    15
    """
    if tree is None:
        return 0
    return 1 + count_entries(tree.left) + count_entries(tree.right)

def big_tree(left, right):
    """Return a tree of elements between left and right.

    >>> big_tree(0, 12)
    Tree(6, Tree(2, Tree(0), Tree(4)), Tree(10, Tree(8), Tree(12)))
    """
    if left > right:
        return None
    split = left + (right - left)//2
    return Tree(split, big_tree(left, split-2), big_tree(split+2, right))

# Memoization
def memo(f):
    cache = {}
    def memoized(n):
        if n not in cache:
            cache[n] = f(n)
        return cache[n]
    return memoized

def fib(n):
    """Compute the nth Fibonacci number.

    >>> fib(35)
    5702887
    """
    if n == 1:
        return 0
    if n == 2:
        return 1
    return fib(n - 2) + fib(n - 1)

@memo
def fib_memo(n):
    """Compute the nth Fibonacci number.

    >>> fib_memo(35)
    5702887
    >>> fib_memo(100)
    218922995834555169026
    """
    if n == 1:
        return 0
    if n == 2:
        return 1
    return fib_memo(n - 2) + fib_memo(n - 1)

# From lecture 23/24: Recursive lists
class Rlist(object):
    """A recursive list consisting of a first element and the rest.

    >>> s = Rlist(1, Rlist(2, Rlist(3)))
    >>> len(s)
    3
    >>> s[0]
    1
    >>> s[1]
    2
    >>> s[2]
    3
    """
    class EmptyList(object):
        def __len__(self):
            return 0
    empty = EmptyList()

    def __init__(self, first, rest=empty):
        self.first = first
        self.rest = rest

    def __repr__(self):
        f = repr(self.first)
        if self.rest is Rlist.empty:
            return 'Rlist({0})'.format(f)
        else:
            return 'Rlist({0}, {1})'.format(f, repr(self.rest))

    def __len__(self):
        return 1 + len(self.rest)

    def __getitem__(self, i):
        if i == 0:
            return self.first
        return self.rest[i - 1]

def extend_rlist(s1, s2):
    """Return a list containing the elements of s1 followed by those
    of s2.

    >>> s = Rlist(1, Rlist(2, Rlist(3)))
    >>> extend_rlist(s.rest, s)
    Rlist(2, Rlist(3, Rlist(1, Rlist(2, Rlist(3)))))
    """
    if s1 is Rlist.empty:
        return s2
    return Rlist(s1.first, extend_rlist(s1.rest, s2))

def map_rlist(s, fn):
    """Return an Rlist resulting from mapping fn over the elements of
    s.

    >>> s = Rlist(1, Rlist(2, Rlist(3)))
    >>> map_rlist(s, lambda x: x * x)
    Rlist(1, Rlist(4, Rlist(9)))
    """
    if s is Rlist.empty:
        return s
    return Rlist(fn(s.first), map_rlist(s.rest, fn))

def filter_rlist(s, fn):
    """Filter the elements of s by predicate fn.

    >>> s = Rlist(1, Rlist(2, Rlist(3)))
    >>> filter_rlist(s, lambda x: x % 2 == 1)
    Rlist(1, Rlist(3))
    """
    if s is Rlist.empty:
        return s
    rest = filter_rlist(s.rest, fn)
    if fn(s.first):
        return Rlist(s.first, rest)
    return rest

# Take 1: Sets as unordered sequences
s = Rlist(1, Rlist(2, Rlist(3))) # A set is an Rlist with no duplicates

def empty(s):
    return s is Rlist.empty

def set_contains(s, v):
    """Return true if set s contains value v as an element.

    >>> set_contains(s, 2)
    True
    >>> set_contains(s, 5)
    False
    """
    if empty(s):
        return False
    if s.first == v:
        return True
    return set_contains(s.rest, v)

def adjoin_set(s, v):
    """Return a set containing all elements of s and element v.

    >>> t = adjoin_set(s, 4)
    >>> t
    Rlist(4, Rlist(1, Rlist(2, Rlist(3))))
    """
    if set_contains(s, v):
        return s
    return Rlist(v, s)

def intersect_set(set1, set2):
    """Return a set containing all elements common to set1 and set2.

    >>> t = adjoin_set(s, 4)
    >>> intersect_set(t, map_rlist(s, lambda x: x*x))
    Rlist(4, Rlist(1))
    """
    return filter_rlist(set1, lambda v: set_contains(set2, v))

def union_set(set1, set2):
    """Return a set containing all elements either in set1 or set2.

    >>> t = adjoin_set(s, 4)
    >>> union_set(t, s)
    Rlist(4, Rlist(1, Rlist(2, Rlist(3))))
    """
    set1_not_set2 = filter_rlist(set1, lambda v: not set_contains(set2, v))
    return extend_rlist(set1_not_set2, set2)