Ordered (search) trees

Trees are often used to implement databases, where Records are saved in the Nodes. (Recall that a Record holds a key and some data.) There is a standard way of building a tree that holds Records:

A binary tree is ordered if all its Nodes have the ordering property: A Node has the ordering property iff

  1. all the objects contained in the Node's left subtree have keys lessthan the key at the Node itself;
  2. all the objects contained in the Node's right subtree have keys greater-than-or-equal-to the key at the Node itself.
An ordered binary tree is sometimes called a binary search tree.

Here is an example ordered tree, where a Record is an integer paired with a string. (The nodes' addresses are a1, etc. The Leaf objects are represented by dots.) 

        a1: (8,"e")
       /     \
a2: (5,"f")   a3: (11,"el")
 /  \            /      \
.   .    a4: (9, "n")  a5: (13,"th")
             / \           / \
            .   .         .   .
Perhaps this tree was created by inserting the records, (8,"e"), (11,"el"), (9,"n"), (5,"f"), and (13,"th"), one by one into an initially empty Leaf tree --- five Node objects and six Leaf objects were constructed and linked together. Perhaps a method named insert was used make each of the five insertions.

Here is one possible specification of insertion: 

/** insert inserts an object in its proper position in an ordered tree
    * @param v - the Record to be inserted
    * @param t - the existing tree
    * @return (the address of) a tree that looks just like  t
    *   except it holds  v  also  */
public BinaryTree insert(Record v, BinaryTree t)
We write these simple schematic equations for the solution: 
insert(v, Leaf ) =  return  Node(v, Leaf(), Leaf());

insert(v, Node(u, l, r) ) = 
            if ( v.getKey() < u.getKey() )
                 { return  Node(u, insert(v, l), r) }
            else { return  Node(u, l, insert(v, r)) }
(Please assume there is a getKey method that extracts the integer key held in a record.)

The first equation says that an insertion into a leaf tree demands that we return (the address of) a node tree that holds v.

The second equation says that an insertion into a nonempty, node tree, requires that we decide whether to insert downwards into the left subtree or downwards into the right subtree.

Insertion into a mutable tree

We can implement the insertion schema in two ways: (i) with a mutable tree or (ii) with an immutable tree. Let's attempt solution (i) first. We do this in three steps:

  1. Starting from the root, search downwards for the Leaf that we will replace with the new record, v.
  2. Once we find the leaf, create a new Node that holds record, v.
  3. Discard the leaf and replace it with the new Node.

To do Step 3, we must use setLeft and setRight methods to alter the existing tree's structure. class Node must include these two methods: 

public class Node extends BinaryTree
{ private Object val;
  private BinaryTree left;
  private BinaryTree right;
    ...

  public void setLeft(BinaryTree new_left)
  { left = new_left; }

  public void setRight(BinaryTree new_right)
  { right = new_right; }
}
Here is the coding for mutable insertion. 
public BinaryTree insert(Record v, BinaryTree t)
{ BinaryTree answer;
  if ( t instanceof Leaf )
     { answer = new Node(v, new Leaf(), new Leaf());  // we make a new Node
                                                      // to hold  v.
  else // t  is a Node:
     { Record u = t.value();
       if ( v.getKey() < u.getKey() ) // remember to properly recode  < 
            { BinaryTree new_left = insert(v, t.left());
              // attach the revised left subtree to  t:
              t.setLeft(new_left);
            }
       else // v's key is  >= u's key :
            { BinaryTree new_right = insert(v, t.right());
              t.setRight(new_right);
            }
       answer = t; 
     }
  return answer;
}
We use recursions to locate the Leaf where the object, v, should be inserted, where we construct a new Node to hold v. This is the only Node that is constructed. We use the setLeft and setRight operations to reset the existing Nodes' branches to the new Node.

Notice that the method cannot insert Record v into tree t, when t is a Leaf! In this case, a new Node must be constructed and must replace t. This is the reason why the method returns the address of the altered tree.

Here is a typical use of insert in an application: 

public class MyDataStructureBasedOnATree
{ private BinTree mytree;   

  // the constructor method builds an empty tree --- a leaf:
  public MyDataStructureBasedOnATree()
  { mytree = new Leaf(); }
   ...

   // the insertRecord method uses the  insert  method we just coded:
   public void insertRecord(Record v)
   { mytree = insert(v, mytree); }
}

Ordered tree lookup is a binary search

Finding an element in an ordered tree can be conducted with a binary-search-like method (which also executes in time order log2n for a tree that holds n values); here is its specification: 

/** find  searches for an an object in an ordered tree 
  * @param k - the key of the object to be found
  * @return the address of the Node within  t  where  k  is found;
  *   if  k  is not found in  t,  return null  */
public Node find(Key k, BinaryTree t)
Here are the schematic equations: 
find(k, Leaf ) = return  null;

find(k, Node(u, l, r) ) =
          if ( k == u.getKey() ) { return the address of this node; }
          if ( k < u.getKey() ) { return find(k, l) }
          if ( k > u.getKey() ) { return find(k, r) }
Here's how we might code the equations in Java: 
public Node find(Key k, BinaryTree t)
{ Node found;  // holds the address of the Node where  k  is found in  t
  if ( t instanceof Leaf )  // have we reached the end of the search?
       { found = null; }    // yes, and we failed to find  k 
  else // t is a Node, so let's ask if we have found  k  here:
       { if ( k == t.value().getKey() )  // NOTE: remember to replace  ==  with the
                                //   correct operation for checking equality
                 { found = t; }
         else if ( k < t.value().getKey() )  // NOTE: remember to replace  <  with the
                                    //  correct operation for checking lessthan
		 // search downwards to the left:
                 { found = find(k, t.left()); }
         else // k > t.value().getKey(), so search downwards to the right:
                 { found = find(k, t.right()); }
       }
  return found;
}
Because the ordered tree has ``sorted'' its values from ``left to right,'' the recursive lookup operates like the binary search algorithm on sorted arrays, searching only that half of the tree where the desired object must reside.

Insertion into an immutable tree (optional material)

Recall from the previous Lecture that it is possible to do a Record insertion into a tree without resetting any links to subtrees! This approach builds a new entire tree that looks just like the tree we started with but it also holds the newly inserted Record: 
public BinaryTree insert(Record v, BinaryTree t)
{ BinaryTree answer;
  if ( t instanceof Leaf )
       // then, build a new node to hold v:
       { answer = new Node(v, new Leaf(), new Leaf()); }
  else // t is a Node:
       { if ( v.getKey() < t.value().getKey() ) // remember to replace  <  with the
                               //  correct operation for checking  <
              // Then, insert  v  into the left subtree and rebuild
              //  the tree from its root value, its new left subtree,
              //  and its existing right subtree:
                 { answer =
                     new Node(t.value(), insert(v, t.left()), t.right());
                 }
         else // v's key is  >= t.value()'s key,  so insert  v  into the right subtree
	      //  and rebuild the tree from its parts:
                 { answer =
                     new Node(t.value(), t.left(), insert(v, t.right()));
                 }
       }
  return answer;
}
Notice that parts of the original tree are reused in the tree we build. For example, given this ordered tree: 
        a1: (8,"e")
         /     \
a2: (5,"f")   a3: (11,"el")
   /  \            /      \
  .   .    a4: (9, "n")  a5: (13,"th")
                / \           / \
               .   .         .   .
inserting the record holding (7,"s") would build this tree as its answer: 
        a8: (8,"e")
         /     \
a7: (5,"f")   a3: the same subtree as above --- it's reused in the answer
   /  \            /      \
  .  a6: (9, "n")
         / \    
        .   .
That is, the invocation, insert(new Record(7,"s"), a1) triggers this instruction: 
answer = new Node(a1.value(), insert(new Record(7,"s"), a1.left()), a1.right());
which computes to 
answer = new Node(8, insert(new Record(7,"s"), a2), a3);
This means that subtree a3 is used, unaltered, in the updated tree.

The recursion, insert(new Record(7,"s"), a2), generates another recursion, 

answer = new Node(a2.value(), a2.left(), insert(new Record(7,"s"), a2.right()));
which computes to 
answer = new Node(5, Leaf(), insert(new Record(7,"s"), Leaf()));

Finally, the insertion of new Record(7,"s") into a leaf causes Node a6 to be constructed, and then the parent nodes, a7 and a8, get constructed.

Although the construction of the answer tree looks more complex than that seen with mutable trees, there is a payback: Remember that the main advantage of immutable trees is that a program can maintain multiple trees that share each others' subtrees and the program can easily implement ``undo'' operations on a tree. (Think about how we might undo the insertion of 7 in the previous example --- we merely revert to the node a1 as the root node; it still exists, unaltered.)

A less pretty method for insertion into a mutable tree (optional material)

Let's return to the coding for inserting into a tree and resetting links.

It is tempting to rewrite its insert method as follows: 

/** insert  inserts object  v  in its proper position in an
 *  ordered tree, t,  by replacing one of  t's  leaves by a Node holding v
 * @param v - the object to be inserted
 * @param t - the tree to be altered  */
public void insert(Object v, BinTree t)
{ ... }
This version does not return the address of the updated tree. With this variant, we will have a problem with this example: 
BinaryTree mytree = new Leaf();
insert("a", mytree);
There is no way that insert can reset mytree, which is a Leaf, to hold a Node that holds "a". This example shows why both codings of insert seen earlier return the address of the updated tree.

But there is one last variant of insertion into mutable trees that need not return as its answer the address of the altered tree. This variant requires that the tree to be altered is known to be a Node: 

/** insertIntoNode  places  u  into its proper position in _Node_ tree  n.
  * @param u - the value to be inserted
  * @param n - the Node tree that is mutated.  */
public void insertIntoNode(Object u, Node n)  // note that  n  must be a Node!
{ if ( u < n.value() )  // insert to the left?
       { if ( n.left() instanceof Leaf )  // insert here?
              { n.setLeft( new Node(u, new Leaf(), new Leaf()) ); }
         else // n is a Node, so descend into its left subtree:
	      { insertIntoNode(u, (Node)(current.left()); }
       }
  else // u >= n.value(), so insert to the right:
       { if ( n.right() instanceof Leaf ) // insert here?
              { n.setRight( new Node(u, new Leaf(), new Leaf()) ); }
         else // n is a Node, so descend into its right subtree:
	      { insertIntoNode(u, (Node)(n.right())); }
       }
}
The correct way of using the above method goes as follows: 
BinTree mytree = ... ;
  ...
// insert object  u  into  mytree:
if ( mytree instance of Leaf )
   { mytree = new Node(u, new Leaf(), new Leaf()); }
else { insert(u, (Node)mytree); }
Notice the castings, (Node), whenever insertIntoNode is used.

Loops and binary trees (optional material)

Trees are normally computed upon with recursive method invocations. But, a loop can compute on a tree if these two conditions hold:
  1. the computation traverses only one path (not all paths!) in the tree
  2. all alterations that the computation makes are along the path traversed
The find method satisfies these two conditions and can be written as a loop. The loop is the usual form of a ``searching loop'': 
public boolean find(Element v, BinTree t)
{ boolean found = false;
  BinTree tree = t;
  while ( !found  &&  !(tree instanceof Leaf) )
        //  invariant:
        //  found==true  implies  v  is in  t
        //  found==false implies, if  v  is in  t, it will be found in  tree
        { if ( v.equals(tree.value() )
               { found = true; }
          else { if ( v.lessthan(tree.value()) )
                      { tree = tree.left(); }
                 else { tree = tree.right(); }
               }
        }
  return found;
}
Each loop iteration causes the value of tree to descend lower and lower into t until the desired object is located.

Finally, we can use the loop pattern to write insertion for a mutable tree, where we know for certain that the tree to be mutated is definitely a Node (and not a Leaf). Here is a not-so-elegant recoding of insertion into a mutable Node tree: 

/** insertIntoNode  places  u  into its proper position in Node tree  n.
  * @param u - the value to be inserted
  * @param n - the Node tree that is mutated.  */
public void insertIntoNode(Object u, Node n)
{ BinTree current = n;
  boolean found_leaf = false;
  while ( !found_leaf )
        { if ( u < current.value() )  // insert to the left?
             { if ( current.left() instanceof Leaf )  // insert here?
                    { found_leaf = true; }
               else { current = current.left(); }
             }
          else // u >= current.value(), so insert to the right:
             { if ( current.right() instanceof Leaf ) // insert here?
                    { found_leaf = true; }
               else { current = current.right(); }
             }
        }
  // we know that we must insert immediately underneath the  current  node:
  if ( u < current.value() )
     { current.setLeft( new Node(u, new Leaf(), new Leaf()) ); }
  else
     { current.setRight( new Node(u, new Leaf(), new Leaf()) ); }
}
The method loops as it searches downwards through the Nodes to locate a Leaf position that can be replaced by value u. After the loop quits, we do just one Node construction and just one link change.