Tree Algorithms

Our textbook recommends a relatively simple representation using Python's tuples (this is a built-in structure that is similar to a list, but immutable). The basic format used is a triple,

(label, leftsubtree,  rightsubtree)

By convention, we will use a representation where if a node doesn't have any children, we will use empty tuples, such as

('C', (), ())

By this convention, a tree that might be represented graphically as

  B
 /
A
 \
  C

would be represented by the recursive structure

('A',
     ('B', (), ()),
     ('C', (), ())
)

although to Python, the whitespace and indentation is not actually important in this context, so this could actually be viewed more streamlined as:

('A', ('B', (), ()), ('C', (), ()) )

As a more complex example, the following tree

   B
  / 
 /      E
A      /
 \    D
  \ / \
   C   F
    \
     G

would be represented by the recursive structure

('A',
     ('B',(),()),
     ('C',
          ('D',
               ('E',(),()),
               ('F',(),())
          ),
          ('G', (), ())
     )
)

which is equally valid in Python as

('A', ('B',(),()), ('C', ('D', ('E',(),()), ('F',(),()) ), ('G', (), ()) ) )

Recursion

While the above tree representation may quickly get unweildy for a human, it turns out that we can write algorithms that work with such a structure quite easily, by using the power of recursion to repeat processes on subtrees. We'll explore several examples of such recursive algorithms.

determining the total number of nodes in a tree

def nodeCount(tree):
    """Return the total number of nodes in the given tree."""
    if tree == ():
        return 0     # base case represents an empty tree
    else:
        return 1 + nodeCount(tree[1]) + nodeCount(tree[2])

determining the maximum height of a tree

def height(tree):
    """Return the number of levels of the tree."""
    if tree == ():
        return 0     # base case represents an empty tree
    else:
        return 1 + max(height(tree[1]), height(tree[2]))

producing a list of labels for all of the leaves of a tree

def leafList(tree):
    """Return list of labels for the leaves of the tree."""
    root,left,right = tree    # unpack the tuple
    if left == right == ():
        return [ root ]       # tree with one node
    else:
        return leafList(tree[1]) + leafList(tree[2])

Michael Goldwasser

Last modified: Wednesday, 21 March 2018