9.3. Working with Trees

I think that I shall never see

A poem as lovely as a tree...

"Trees," [Alfred] Joyce Kilmer

Trees in computer science are a relatively intuitive concept (except that they are usually drawn with the "root" at the top and the "leaves" at the bottom). This is because we are familiar with so many kinds of hierarchical data in everyday life, from the family tree to the corporate organization chart to the directory structures on our hard drives.

The terminology of trees is rich but easy to understand. Any item in a tree is a node; the first or topmost node is the root. A node may have descendants that are below it, and the immediate descendants are called children. Conversely, a node may also have a parent (only one) and ancestors. A node with no child nodes is called a leaf. A subtree consists of a node and all its descendants. To travel through a tree (for example, to print it out) is called traversing the tree.

We will look mostly at binary trees, though in practice a node can have any number of children. We will see how to create a tree, populate it, and traverse it; and we will look at a few real-life tasks that use trees.

We will mention here that in many languages such as C or Pascal, trees would be implemented using true address pointers. But in Ruby (as in Java, for instance), we don't use pointers; object references work just as well or better.

9.3.1. Implementing a Binary Tree

There is more than one way to implement a binary tree in Ruby. For example, we could use an array to store the values. Here we use a more traditional approach, coding much as we would in C, except that pointers are replaced with object references.

What is required to describe a binary tree? Well, each node needs an attribute of some kind for storing a piece of data. Each node also needs a pair of attributes for referring to the left and right subtrees under that node.

We also need a way to insert into the tree and a way of getting information out of the tree. A pair of methods will serve these purposes.

The first tree we'll look at implements these methods in a slightly unorthodox way. Then we will expand on the TRee class in later examples.

A tree is in a sense defined by its insertion algorithm and by how it is traversed. In this first example (see Listing 9.1), we define an insert method that inserts in a breadth-first fashionthat is, top to bottom and left to right. This guarantees that the tree grows in depth relatively slowly and that the tree is always balanced. Corresponding to the insert method, the traverse iterator will iterate over the tree in the same breadth-first order.

Listing 9.1. Breadth-First Insertion and Traversal in a Tree

class Tree

attr_accessor :left
attr_accessor :right
attr_accessor :data

def initialize(x=nil)
  @left = nil
  @right = nil
  @data = x
end

def insert(x)
  list = []
  if @data == nil
    @data = x
  elsif @left == nil
    @left = Tree.new(x)
  elsif @right == nil
    @right = Tree.new(x)
  else
    list << @left
    list << @right
    loop do
      node = list.shift
      if node.left == nil
        node.insert(x)
        break
      else
        list << node.left
      end
      if node.right == nil
        node.insert(x)
        break
      else
        list << node.right
      end
    end
  end
end

def traverse()
  list = []
  yield @data
  list << @left if @left != nil
  list << @right if @right != nil
  loop do
    break if list.empty?
    node = list.shift
    yield node.data
    list << node.left if node.left != nil
    list << node.right if node.right != nil
  enä
end

end


items = [1, 2, 3, 4, 5, 6, 7]

tree = Tree.new

items.each {|x| tree.insert(x)}

tree.traverse {|x| print "#{x} "}
print "\n"

# Prints "1 2 3 4 5 6 7 "

This kind of tree, as defined by its insertion and traversal algorithms, is not especially interesting. It does serve as an introduction and something on which we can build.

9.3.2. Sorting Using a Binary Tree

For random data, a binary tree is a good way to sort. (Although in the case of already sorted data, it degenerates into a simple linked list.) The reason, of course, is that with each comparison, we are eliminating half the remaining alternatives as to where we should place a new node.

Although it might be fairly rare to do this nowadays, it can't hurt to know how to do it. The code in Listing 9.2 builds on the previous example.

Listing 9.2. Sorting with a Binary Tree

class Tree

  # Assumes definitions from
  # previous example...

  def insert(x)
    if @data == nil
      @data = x
    elsif x <= @data
      if @left == nil
        @left = Tree.new x
      else
        @left.insert x
      end
    else
      if @right == nil
        @right = Tree.new x
      else
        @right.insert x
      end
    end
  end

  def inorder()
    @left.inorder {|y| yield y} if @left != nil
    yield @data
    @right.inorder {|y| yield y} if @right != nil
  end

  def preorder()
    yield @data
    @left.preorder {|y| yield y} if @left != nil
    @right.preorder {|y| yield y} if @right != nil
  end

  def postorder()
    @left.postorder {|y| yield y} if @left != nil
    @right.postorder {|y| yield y} if @right != nil
    yield @data
  end
end

items = [50, 20, 80, 10, 30, 70, 90, 5, 14,
         28, 41, 66, 75, 88, 96]

tree = Tree.new

items.each {|x| tree.insert(x)}

tree.inorder {|x| print x, " "}
print "\n"
tree.preorder {|x| print x, " "}
print "\n"
tree.postorder {|x| print x, " "}
print "\n"

# Output:
# 5 10 14 20 28 30 41 50 66 70 75 80 88 90 96
# 50 20 10 5 14 30 28 41 80 70 66 75 90 88 96
# 5 14 10 28 41 30 20 66 75 70 88 96 90 80 50    print "\n"

9.3.3. Using a Binary Tree As a Lookup Table

Suppose we have a tree already sorted. Traditionally this has made a good lookup table; for example, a balanced tree of a million items would take no more than 20 comparisons (the depth of the tree or log base 2 of the number of nodes) to find a specific node. For this to be useful, we assume that the data for each node is not just a single value but has a key value and other information associated with it.

In most, if not all situations, a hash or even an external database table will be preferable. But we present this code to you anyhow:

class Tree

  # Assumes definitions
  # from previous example...

  def search(x)
    if self.data == x
      return self
    elsif x < self.data
      return left ? left.search(x) : nil
    else
      return right ? right.search(x) : nil
    end
  end

end

keys = [50, 20, 80, 10, 30, 70, 90, 5, 14,
        28, 41, 66, 75, 88, 96]

tree = Tree.new

keys.each {|x| tree.insert(x)}

s1 = tree.search(75)   # Returns a reference to the node
                       # containing 75...

s2 = tree.search(100)  # Returns nil (not found)

9.3.4. Converting a Tree to a String or Array

The same old tricks that allow us to traverse a tree will allow us to convert it to a string or array if we want. Here we assume an inorder traversal, though any other kind could be used:

class Tree

  # Assumes definitions from
  # previous example...

  def to_s
    "[" +
    if left then left.to_s + "," else "" end +
    data.inspect +
    if right then "," + right.to_s else "" end + "]"
  end

  def to_a
    temp = []
    temp += left.to_a if left
    temp << data
    temp += right.to_a if right
    temp
  end

end

items = %w[bongo grimace monoid jewel plover nexus synergy]

tree = Tree.new
items.each {|x| tree.insert x}

str = tree.to_a * ","
# str is now "bongo,grimace,jewel,monoid,nexus,plover,synergy"
arr = tree.to_a
# arr is now:
# ["bongo",["grimace",[["jewel"],"monoid",[["nexus"],"plover",
#  ["synergy"]]]]]

Note that the resulting array is as deeply nested as the depth of the tree from which it came. You can, of course, use flatten to produce a non-nested array.