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We have seen that a set of integers can be stored in an array in ascending order, and that binary search allows lookup in the array in time O(log(n)) where n is the size of the set. But we also noticed that using an array does not appear to allow values to be inserted or removed efficiently, and we set a goal of finding a way to achieve lookup, insertion and removal all in time O(log(n)).
We now turn to an idea, binary search trees, that, after some thought, will achieve that goal.
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Tree T is a binary search tree if the following
is true for every node v in T.
If v has item k, then all nodes in the left subtree of v have items that are less than k, and all nodes in the right subtree of v have items that are greater than k. Graphically: 
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For example, the following is a binary search tree.
A binary search tree can have any shape, as long as it obeys the ordering requirement. The following is a binary search tree. (The left subtree of each node is an empty tree.)
A binary tree represents the set of all integers that occur in its nodes. An empty tree represents an empty set.
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