Trees

In recursion.html#exprtrees we looked at some simple tree structures used for expressing "hierarchical" information: expressions:

2*(3+5)

         *
       /   \
      2     +
          /   \
         3     5

• data
• a left subtree
• a right subtree

The code might look like this (from bintree.cs)

    class treenode {
	private T data_;
	private treenode left_, right_;
	public treenode(T d, treenode l, treenode r) {data_ = d; left_ = l; right_ = r;}
	public T data() {return data_;}
	public treenode left()  {return left_;}
	public treenode right() {return right_;}
    }

A binary tree is ordered if there is an ordering relation on the data and, for each node, every data value appearing in the left subtree is less than the node's own data, and every data value appearing in the right subtree is greater than the node's own data.

Quite often, eg for expression trees, no ordering is involved.

The root node is at the top of the tree. Nodes with no subtrees (ie in which the subtree fields are null) are leaf nodes. Other nodes are interior nodes.

The depth of a node is how far it is, in node-to-node links, from the root. Level n of a tree consists of all nodes with depth n.

Examples:

• Abstract expression trees
• Parse trees
• Ancestry trees
• Decision trees (Bailey p 288)

None of these is ordered!

Tree Traversal

To traverse a tree is to visit each node, perhaps printing out the data values or perhaps taking some other action. Traversal methods are usually recursive, and usually the initial call is on the tree's root.

Three common forms of traversal are preorder, inorder, or postorder, depending on whether a node's data is visited before, between or after the left and right subtrees are visited. All of these are considered to be depth-first traversal. Here is a sample inorder traversal

Demo:

    traverse the expression tree for 4+x*(y+1)

        inorder

        postorder (cf Jan Łukasiewicz)

    traverse the following tree from inttree.cs:

Tree Building

How do we build trees? One way is from the bottom up; that is much easier of the class treenode is public. We could do this (assuming treenode is equivalent to treenode):

    treenode n1 = new treenode("a", null, null);
    treenode n2 = new treenode("3", null, null);
    treenode n3 = new treenode("*", n1, n2);
    treenode n4 = new treenode("1", null, null);
    treenode root = new treenode("+", n3, n4);

What does this build?

We can build ordered trees by inserting in order; see inttree.cs:

    public void insert(int val) {
	if (root_ == null) {
		root_ = new treenode(val, null, null);
		return;
	}
	rinsert(val, root_);
    }

    // recursive insert: does not get called with t==null
    private void rinsert(int val, treenode t) {
	if (val == t.data() ) return;	// should we do something else?
	if (val < t.data()) {
	    if (t.left() == null) {
		t.setleft(new treenode(val, null, null));
	    } else {
		rinsert(val, t.left());
	    }
	} else { // val > t.data()
	    if (t.right() == null) {
		t.setright(new treenode(val, null, null));
	    } else {
		rinsert(val, t.right());
	    }
	}
    }

If a treenode is null, we cannot modify it; we can only modify the parent. Bailey addresses this in some contexts by including a pointer to the parent node, but that won't help us here.

Note that the tree we get by inserting values depends on the order of insertion. 

In-class demos: 

• try inserting the values in order
• what orders lead to the same tree?
• convert to preorder traversal

Lab 4: use this idea to implement a StrList that has O(log(n)) times for both search and insert.

n-ary trees and ordered n-ary trees