Binary Search Trees
A binary tree is like a linked list, but instead of a single “next” link, it has two “left
and right”. Everything smaller than, or equal to data gets assigned to left and
everything greater than data gets assigned to right.
Consider adding the numbers: 50,30,75,15,26,64,55,70 to a binary tree. It would
end up looking like this:
50
30
75
15
64
26
55
70
With a simple binary search tree the efficiency of retrieving a given item depends
heavily on the order the items were inserted. If a tree remains balanced searches
would run at O(Log n). For now, all you need to know is:

A balanced tree is faster

Only as many comparisons as the tree has levels, and there will be O(Log n)
levels
Traversing a Tree
The binary tree is a natural fit for recursive logic – since each node is a subset
structure of the larger tree. Consider a print algorithm:
PrintTree( node ){
if current.left !=null
PrintTree( current.left )
Print Current
if current.right !=null
PrintTree(current.right)
}
This algorithm will print the list in order since it traverses down the left while left
!=null. When left is null it prints the element (15), then Print the right node. Both
elements are null there, so it just prints current. The previous recursive call
continues and outputs current (30) then (50) and then moves on to print the right
side of the tree.
A search algorithm therefore would be:
SearchTree( node, target){
If target=node
Return node
Else if target<=node and node.left!=null
SearchTree( node.left, target)
Else if target>node and node.right!=null
SearchTree( node.right, target)
Else
Handle node not found exception
}
In this example we check to see if we are looking at the target node. If we are, we
return it, if we’re not we follow the rules of the tree to find it: Search the left subtree if target is less than, or equal to current, and the right sub-tree if target is
greater than current. Finally, let’s look at an add algorithm:
AddNode(current, Integer temp){
If start=null
Start=temp
Else If current >= temp
If current.left=null
Current.left = new Node(temp)
Else
AddNode(current.left, temp)
Else If current < temp
If current.right=null
Current.right = new Node(temp)
Else
AddNode(current.right, temp)
This follows similar logic to the search algorithm. Navigate current to the correct end
node (through sub-trees until there are no more sub-trees) and add temp to the
correct pointer.
Exercise
Show a binary tree after these numbers are added:
8,5,7,6,3,8,9,2,1,0,2
Exercise
Implement a Binary Tree of type Comparable which has the methods described
above. Use it to store a dozen Widgets.