Section 9.4. Working with Graphs

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.