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Trees

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- A tree is a set of nodes.
- A tree may be empty (i.e., contain no nodes).
- If not empty, there is a distinguished node, r, called the root and zero or more non-empty subtrees T1, T2, T3,….Tk, each of whose roots is connected by a directed edge from r.
- Trees are recursive in their definition and, therefore, in their implementation.

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- Tree terminology takes its terms from both nature and genealogy.
- A node directly below the root, r, of a subtree is a child of r, and r is called its parent.
- All children with the same parent are called siblings.
- A node with one or more children is called an internal node.
- A node with no children is called a leaf or external node.

- A path in a tree is a sequence of nodes, (N1, N2, … Nk) such that Ni is the parent of Ni+1 for 1 <= i <= k.
- The length of this path is the number of edges encountered (k – 1).
- If there is a path from node N1 to N2, then N1 is an ancestor of N2 and N2 is a descendant of N1.

- The depth of a node is the length of the path from the root to the node.
- The height of a node is the length of the longest path from the node to a leaf.
- The depth of a tree is the depth of its deepest leaf.
- The height of a tree is the height of the root.
- True or False – The height of a tree and the depth of a tree always have the same value.

- First attempt - each tree node contains
- The data being stored
- We assume that the objects contained in the nodes support all necessary operations required by the tree.

- Links to all of its children

- The data being stored
- Problem: A tree node can have an indeterminate number of children. So how many links do we define in the node?

- Since we can’t know how many children a node can have, we can’t create a static data structure -- we need a dynamic one.
- Each node will contain
- The data which supports all necessary operations
- A link to its first child
- A link to a sibling

- To be supplied in class

- Traversing a tree means starting at the root and visiting each node in the tree in some orderly fashion. “visit” is a generic term that means “perform whatever operation is applicable”.
- “Visiting” might mean
- Print data stored in the tree
- Check for a particular data item
- Almost anything

- Start at the root.
- Visit all the root’s children.
- Then visit all the root’s grand-children.
- Then visit all the roots great-grand-children, and so on.
- This traversal goes down by levels.
- A queue can be used to implement this algorithm.

Create a queue, Q, to hold tree nodes

Q.enqueue (the root)

while (the queue is not empty)Node N = Q.dequeue( )for each child, X, of N Q.enqueue (X)

- The order in which the nodes are dequeued is the BF traversal order.

- Start at the root.
- Choose a child to visit; remember those not chosen
- Visit all of that child’s children.
- Visit all of that child’s children’s children, and so on.
- Repeat until all paths have been traversed
- This traversal goes down a path until the end, then comes back and does the next path.
- A stack can be used to implement this algorithm.

Create a stack, S, to hold tree nodes

S.push (the root)

While (the stack is not empty)Node N = S.pop ( )for each child, X, of NS.push (X)

The order in which the nodes are popped is the DF traversal order.

- What is the asymptotic performance of breadth-first and depth-first traversals on a general tree?

- A binary tree is a tree in which each node may have at most two children and the children are designated as left and right.
- A full binary tree is one in which each node has either two children or is a leaf.
- A perfect binary tree is a full binary tree in which all leaves are at the same level.

A binary tree?

A full binary tree?

A binary tree?

A full binary tree?

A perfect binary tree?

- Because nodes in binary trees have at most two children (left and right), we can write specialized versions of DF traversal. These are called
- In-order traversal
- Pre-order traversal
- Post-order traversal

- At each node
- visit my left child first
- visit me
- visit my right child last

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void inOrderTraversal(Node *nodePtr) {

if (nodePtr != NULL) {

inOrderTraversal(nodePtr->leftPtr);

cout << nodePtr->data << endl;

inOrderTraversal(nodePtr->rightPtr);

}

}

- At each node
- visit me first
- visit my left child next
- visit my right child last

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void preOrderTraversal(Node *nodePtr) {

if (nodePtr != NULL) {

cout << nodePtr->data << endl;

preOrderTraversal(nodePtr->leftPtr);

preOrderTraversal(nodePtr->rightPtr);

}

}

- At each node
- visit my left child first
- visit my right child next
- visit me last

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void postOrderTraversal(Node *nodePtr) {

if (nodePtr != NULL) {

postOrderTraversal(nodePtr->leftPtr);

postOrderTraversal(nodePtr->rightPtr);

cout << nodePtr->data << endl;

}

}

- Recall that the data stored in the nodes supports all necessary operators. We’ll refer to it as a “value” for our examples.
- Typical operations:
- Create an empty tree
- Insert a new value
- Search for a value
- Remove a value
- Destroy the tree

- Set the pointer to the root node equal to NULL.

- The first value goes in the root node.
- What about the second value?
- What about subsequent values?

- Since the tree has no properties which dictate where the values should be stored, we are at liberty to choose our own algorithm for storing the data.

- Since there is no rhyme or reason to where the values are stored, we must search the entire tree using a BF or DF traversal.

- Once again, since the values are not stored in any special way, we have lots of choices. Example:
- First, find the value via BF or DF traversal.
- Second, replace it with one of its descendants (if there are any).

- We have to be careful of the order in which nodes are destroyed (deallocated).
- We have to destroy the children first, and the parent last (because the parent points to the children).
- Which traversal (in-order, pre-order, or post-order) would be best for this algorithm?

- Binary trees can be made more useful if we dictate the manner in which values are stored.
- When selecting where to insert new values, we could follow this rule:
- “left is less”
- “right is more”

- Note that this assumes no duplicate nodes (i.e., data).

- A binary tree with the additional property that at each node,
- the value in the node’s left child is smaller than the value in the node itself, and
- the value in the node’s right child is larger than the value in the node itself.

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- Searching for the value X, given a pointer to the root
- If the value in the root matches, we’re done.
- If X is smaller than the value in the root, look in the root’s left subtree.
- If X is larger than the value in the root, look in the root’s right subtree.

- A recursive routine – what’s the base case?

- To insert value X in a BST
- Proceed as if searching for X.
- When the search fails, create a new node to contain X and attach it to the tree at that node.

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- Non-trivial
- Three separate cases:
- node is a leaf (has no children)
- node has a single child
- node has two children

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- The fact that the values in the nodes are in a special order doesn’t help.
- We still have to destroy each child before destroying the parent.
- Which traversal must we use?

- What is the asymptotic performance of:
- insert
- search
- remove

- Is the performance of insert, search, and remove for a BST improved over that for a plain binary tree?
- If so, why?
- If not, why not?