Solutions: You can find the file with solutions for all questions here.

Trees

Question 1: Search

Write a function search that returns the Tree, whose entry is the given value if it exists and None if it does not. You can assume all entries are unique.

def search(t, value):
    """Searches for and returns the Tree whose entry is equal to value if
    it exists and None if it does not. Assume unique entries.

    >>> t = Tree(1, [Tree(3, [Tree(5)]), Tree(7)])
    >>> search(t, 10)
    >>> search(t, 5)
    Tree(5)
    >>> search(t, 1)
    Tree(1, [Tree(3, [Tree(5)]), Tree(7)])
    """
    if t.entry == value:
        return t
    for branch in t.branches:
        result = search(branch, value)
        if result is not None:
            return result
    return

Use OK to test your code:

python3 ok -q search

Question 2: Tree Map

Define the function tree_map, which takes in a tree and a one-argument function as arguments and returns a new tree which is the result of mapping the function over the entries of the input tree.

def tree_map(fn, t):
    """Maps the function fn over the entries of t and returns the
    result in a new tree.
    >>> numbers = Tree(1,
    ...                [Tree(2,
    ...                      [Tree(3),
    ...                       Tree(4)]),
    ...                 Tree(5,
    ...                      [Tree(6,
    ...                            [Tree(7)]),
    ...                       Tree(8)])])
    >>> print(tree_map(lambda x: 2**x, numbers))
    2
      4
        8
        16
      32
        64
          128
        256
    >>> print(numbers)
    1
      2
        3
        4
      5
        6
          7
        8
    """
    if t.is_leaf():
        return Tree(fn(t.entry), [])
    mapped_subtrees = [tree_map(fn, b) for b in t.branches]
    return Tree(fn(t.entry), mapped_subtrees)

# Alternate solution
def tree_map(fn, t):
    return Tree(fn(t.entry), [tree_map(fn, b) for b in t.branches])

Video walkthrough: https://youtu.be/pbMeCRUU7yw?t=26m47s

Use OK to test your code:

python3 ok -q tree_map

Question 3: Add Leaves

Implement add_d_leaves, a function that takes in a Tree instance t and mutates it so that at each depth d in the tree, d leaves with labels v are added to each node at that depth. For example, we want to add 1 leaf with v in it to each node at depth 1, 2 leaves to each node at depth 2, and so on.

Recall that the depth of a node is the number of edges from that node to the root, so the depth of the root is 0. The leaves should be added to the end of the list of branches.

def add_d_leaves(t, v):
    """Add d leaves containing v to each node at every depth d.

    >>> t1 = Tree(1, [Tree(3)])
    >>> add_d_leaves(t1, 4)
    >>> t1
    Tree(1, [Tree(3, [Tree(4)])])
    >>> t2 = Tree(2, [Tree(5), Tree(6)])
    >>> t3 = Tree(3, [t1, Tree(0), t2])
    >>> add_d_leaves(t3, 10)
    >>> print(t3)
    3
      1
        3
          4
            10
            10
            10
          10
          10
        10
      0
        10
      2
        5
          10
          10
        6
          10
          10
        10
    """
    def add_leaves(t, d):
        for b in t.branches:
            add_leaves(b, d + 1)
        t.branches.extend([Tree(v) for _ in range(d)])
    add_leaves(t, 0)

Use OK to test your code:

python3 ok -q add_d_leaves

Question 4: Long Paths

Implement long_paths, which returns a list of all paths in a tree with length at least n. A path in a tree is a linked list of node values that starts with the root and ends at a leaf. Each subsequent element must be from a branch of the previous value's node. The length of a path is the number of edges in the path (i.e. one less than the number of nodes in the path). Paths are listed in branch order i.e. from left to right.

def long_paths(tree, n):
    """Return a list all paths in tree with length at least n.

    >>> t = Tree(3, [Tree(4), Tree(4), Tree(5)])
    >>> left = Tree(1, [Tree(2), t])
    >>> mid = Tree(6, [Tree(7, [Tree(8)]), Tree(9)])
    >>> right = Tree(11, [Tree(12)])
    >>> whole = Tree(0, [left, Tree(13), mid, right])
    >>> for path in long_paths(whole, 2):
    ...     print_link(path)
    ...
    <0 1 2>
    <0 1 3 4>
    <0 1 3 4>
    <0 1 3 5>
    <0 6 7 8>
    <0 6 9>
    <0 11 12>
    >>> for path in long_paths(whole, 3):
    ...     print_link(path)
    ...
    <0 1 3 4>
    <0 1 3 4>
    <0 1 3 5>
    <0 6 7 8>
    >>> long_paths(whole, 4)
    []
    """
    paths = []
    if n <= 0 and not tree.branches:
        paths.append(Link(tree.entry))
    for b in tree.branches:
        for path in long_paths(b, n - 1):
            paths.append(Link(tree.entry, path))
    return paths

Use OK to test your code:

python3 ok -q long_paths

Optional Question

Question 5: Partial Tree

The sequence_to_tree function takes a sorted linked list and converts it into a balanced Tree in which every sub-tree has at most two branches. For every subtree with 2 branches, all nodes in the left branch are smaller than the root of the subtree and all nodes in the right branch are larger.

A Tree is balanced if

  • It is a leaf, or
  • It has exactly one branch that is a leaf, or
  • It has two branches and the number of nodes in its first branch differs from the number of nodes in its second branch by at most 1, and both branches are also balanced.

In order to write sequence_to_tree, implement partial_tree(s, n), which converts the first n elements of the sorted linked list s into a balanced Tree. The return value is a two-element tuple: the resulting balanced tree; and the rest of the linked list.

Hint: This function requires two recursive calls. The first call builds a left branch out of the first left_size elements of s; Then, the next element is used as the entry of the returned tree. Finally, the second recursive call builds the right branch out of the next right_size elements. In total, (left_size + 1 + right_size) = n, where 1 is for the entry:

def partial_tree(s, n):
    """Return a balanced tree of the first n elements of Link s, along with
    the rest of s.

    Examples of balanced trees:

    Tree(1)                      # leaf
    Tree(1, [Tree(2)])           # one branch is a leaf
    Tree(1, [Tree(2), Tree(3)])  # two branches with one node each

    Examples of unbalanced trees:

    Tree(1, [Tree(2, [Tree(3)])])            # one branch not a leaf
    Tree(1, [Tree(2),                        # Mismatch: branch with 1 node
             Tree(3, [Tree(4, [Tree(5)])])]) #        vs branch with 3 nodes

    >>> s = Link(1, Link(2, Link(3, Link(4, Link(5)))))
    >>> partial_tree(s, 3)
    (Tree(2, [Tree(1), Tree(3)]), Link(4, Link(5)))
    >>> t = Link(-2, Link(-1, Link(0, s)))
    >>> partial_tree(t, 7)[0]
    Tree(1, [Tree(-1, [Tree(-2), Tree(0)]), Tree(3, [Tree(2), Tree(4)])])
    >>> partial_tree(t, 7)[1]
    Link(5)
    """
    if n == 1:
        return (Tree(s.first), s.rest)
    elif n == 2:
        return (Tree(s.first, [Tree(s.rest.first)]), s.rest.rest)
    else:
        left_size = (n-1)//2
        right_size = n - left_size - 1
        left, rest = partial_tree(s, left_size)
        entry, rest = rest.first, rest.rest
        right, rest = partial_tree(rest, right_size)
        return Tree(entry, [left, right]), rest

def sequence_to_tree(s):
    """Return a balanced tree containing the elements of sorted Link s.

    Note: this implementation is complete, but the definition of partial_tree
    above is not complete.

    >>> sequence_to_tree(Link(1, Link(2, Link(3))))
    Tree(2, [Tree(1), Tree(3)])
    >>> elements = Link(1, Link(2, Link(3, Link(4, Link(5, Link(6, Link(7)))))))
    >>> sequence_to_tree(elements)
    Tree(4, [Tree(2, [Tree(1), Tree(3)]), Tree(6, [Tree(5), Tree(7)])])
    """
    return partial_tree(s, len(s))[0]

Use OK to test your code:

python3 ok -q partial_tree