Working with Trees

Working with Trees

I think that I shall never see

A poem as lovely as a tree….

[Alfred] Joyce Kilmer, "Trees"

A tree in computer science is a relatively intuitive concept (except that it is 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 lifefrom the family tree, to the corporate org 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, although in practice a node can have any number of children. You will see how to create a tree, populate it, and traverse it. Also, we will look at a few real-life tasks that use trees.

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

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 in order 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 will implement these methods in a slightly unorthodox way. 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, shown in Listing 3.16, we define an insert method that inserts in a breadth-first fashion (that is, top to bottom and left to right). This guarantees that the tree grows in depth relatively slowly and is always balanced. Corresponding to the insert method, the traverse iterator will iterate over the tree in the same breadth-first order.

Listing 3.16 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
  end
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. However, it does serve as an introduction and something on which we can build.

Sorting Using a Binary Tree

For random data, using 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 sort using a binary tree nowadays, it can't hurt to know how. The code in Listing 3.17 builds on the previous example.

Listing 3.17 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"

Using a Binary Tree As a Lookup Table

Suppose we have a tree already sorted. Traditionally, this has made for 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. However, we present this code to you anyhow (see Listing 3.18).

Listing 3.18 Searching a Binary Tree

class Tree

  # Assumes definitions
  # from previous example...

  def search(x)
    if self.data == x
      return self
    else
      ltree = left != nil ? left.search(x) : nil
      return ltree if ltree != nil
      rtree = right != nil ? right.search(x) : nil
      return rtree if rtree != nil
    end
    nil
  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)

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 wish, as shown in Listing 3.19. Here, we assume an inorder traversal, although any other kind could be used.

Listing 3.19 Converting a Tree to a String or Array

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_s * ","
# 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.

Storing an Infix Expression in a Tree

Here is another little contrived problem illustrating how a binary tree might be used (see Listing 3.20). We are given a prefix arithmetic expression and want to store it in standard infix form in a tree. (This is not completely unrealistic because the Ruby interpreter itself stores expressions in a tree structure, although it is a couple of orders of magnitude greater in complexity.)

We define a "standalone" method called addnode that will add a node to a tree in the proper place. The result will be a tree in which every leaf is an operand and every non-leaf node is an operator. We also define a new Tree method called infix , which will traverse the tree in order and act as an iterator. One twist is that it adds in parentheses as it goes, because prefix form is "parenthesis free" but infix form is not. The output would look more elegant if only necessary parentheses were added, but we added them indiscriminately to simplify the code.

Listing 3.20 Storing an Infix Expression in a Tree

class Tree

  # Assumes definitions from
  # previous example...

  def infix()
    if @left != nil
      flag = %w[* / + -].include? @left.data
      yield "(" if flag
      @left.infix { y yield y}
      yield ")" if flag
    end
    yield @data
    if @right != nil
      flag = %w[* / + -].include? @right.data
      yield "(" if flag
      @right.infix { y yield y}  if @right != nil
      yield ")" if flag
    end
  end

end

def addnode(nodes)
  node = nodes.shift
  tree = Tree.new node
  if %w[* / + -].include? node
    tree.left  = addnode nodes
    tree.right = addnode nodes
  end
  tree
end


prefix = %w[ * + 32 * 21 45 - 72 + 23 11 ]
tree = addnode prefix

str = ""
tree.infix { x str += x}
# str is now "(32+(21*45))*(72-(23+11))"

Additional Notes on Trees

We'll mention a few more notes on trees here. First of all, a tree is a special case of a graph (as you will see shortly); in fact, it is a directed acyclic graph (DAG). Therefore, you can learn more about trees by researching graph algorithms in general.

There is no reason that a tree should necessarily be binary; this is a convenient simplification that frequently makes sense. However, it is conceivable to define a multiway tree in which each node is not limited to two children but may have an arbitrary number. In such a case, you would likely want to represent the child node pointers as an array of object references.

A B-tree is a specialized form of multiway tree. It is an improvement over a binary tree in that it is always balanced (that is, its depth is minimal), whereas a binary tree in a degenerate case can have a depth that is equal to the number of nodes it has. There is plenty of information on the Web and in textbooks if you need to learn about B-trees. Also, the principles we've applied to ordinary binary trees can be extended to B-trees as well.

A red-black tree is a specialized form of binary tree in which each node has a color (red or black) associated with it. In addition, each node has a pointer back to its parent (meaning that it is arguably not a tree at all because it isn't truly acyclic). A red-black tree maintains its balance through rotations of its nodes; that is, if one part of the tree starts to get too deep, the nodes can be rearranged so that depth is minimized (and in-order traversal ordering is preserved). The extra information in each node aids in performing these rotations.

Another tree that maintains its balance in spite of additions and deletions is the AVL tree. This structure is named for its discoverers, the two Russian researchers Adel'son-Vel'skii and Landis. An AVL tree is a binary tree that uses slightly more sophisticated insertion and deletion algorithms to keep the tree balanced. It performs rotation of subtrees similar to that done for red-black trees.

All these and more are potentially useful tree structures. If you need more information, search the Web or consult any book on advanced algorithms.