Trees: basic definitions and terminology

Contrary to arrays, stacks, queues and sequences all of which are one-

dimensional data structures, trees are two-dimensional data structures with

hierarchical relationship between data items.

Definition 1 A tree is a non-empty collection of vertices (nodes) and edges that satisfy certain requirements.

Definition 2 A path in a tree is a list of distinct vertices in which successive vertices are connected by edges in the tree.

One node in the tree is designated as the root. Each tree has exactly one

path between the root and each of the other nodes. If there is more than one

path between the root and some node, or no path at all, we have a graph.

Definition 3 A set of trees is called a forest.

Definition 4 (Recursive definition) A tree is either a single node or a root node connected to a forest.

Example of a tree: root

siblings

subtree

internal nodes

external nodes, or leaves

More definitions

Definition 5 An ordered tree is a tree in which the order of children is specified.

Definition 6 A level (depth) of a node in the number of nodes on the path from that node to the root.

Definition 7 The height (maximum distance) of a tree is the maximum level among all of the nodes in the tree.

Definition 8 The path length of a tree is the sum of the levels of all the nodes in the tree.

Definition 9 A tree where each node has a specific number of children appearing in a specific order is call a multiway tree. The simplest type of a multiway tree is the binary tree. Each node in a binary tree has exactly two children one of which is designated as a left child, and the other is designated as a right child.

Definition 10 (Recursive definition) A binary tree is either an external node, or an internal node and two binary trees.

Example of a binary tree

root

left child right child

one or both children

might be external nodes

special external nodes with

no name and no data associated

with them

More binary trees examples

1. Binary tree for representing arithmetic expressions. The underlying hierarchical relationship is that of an arithmetic operator and its two operands.

Arithmetic expression in an infix form: (A - B) + C * (E / F)

+

- *

A B C /

E F

Note that a post-order traversal of this tree (i.e. visiting the left subtree first, right

subtree next, and finally the root) returns the postfix form of the arithmetic

expression, while the pre-order traversal (root is visited first, then the left subtree,

then the right subtree) returns the prefix form of the arithmetic expression.

2. Binary tree with a heap property. The underlying hierarchical relationship suggests that the datum in each node is greater than or equal to the data in its left and right subtrees.

87

84 63

68 79 12

32 67 6 10

8 9

3. Binary tree with an ordering property. The underlying hierarchical relationship suggests that the datum in each node is greater than the data in its left subtree, and less than or equal to the data in its right subtrees.

87

84 103

68 86 90 109

32 74 88 97

70 80

4. Decision trees. The underlying hierarchical relationship depends on the nature of the domain represented by the binary tree. For example, consider a domain that consists of the following statements (from J.Ignizio “Intro to ES”): If the plane’s engine is propeller, then the plane is C130. If the plane’s engine is jet and the wing position is low, then the plane is B747. If the plane’s engine is jet and the wing position is high and no bulges are seen, then

the plane is C5A If the plane’s engine is jet and the wing position is high and bulges are aft of wing, then

plane is C141 .

The following decision tree can be generated from these rules:

Engine type Jet Propeller

Wing Position C130 Low High

B747 Bulges None Aft Wing

C5A C141

Properties of binary trees

1. The number of external nodes is 1 more than the number of internal nodes. It is easy to see this if we start removing external nodes with their internal parent, one pair at a time (assume that a method removeAboveExternal(n) does this). At the end of this process, only the root with its two external children will remain.

2. The number of external nodes is at least h + 1, where h is the height of the tree, and at most 2h . The later holds for a full binary tree, which is a tree where internal nodes completely fill every level.

3. The number of internal nodes is at least h and at most 2h - 1.

4. The total number of nodes in a binary tree is at least 2*h + 1 and at most 2h+1 - 1.

5. The height, h, of a binary tree with n nodes is at least log n+1 and at most n.

6. A binary tree with n nodes has exactly n - 1 edges.

Full binary trees and complete binary trees

Here is an example of a full binary tree:

1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

A complete binary tree is a full binary tree where the internal nodes on the

bottom level all appear to the left of the external nodes on that level. Here is

an example of a complete binary tree:

1

2 3

4 5 6

Properties of binary trees (cont.)

The following property holds for a complete binary tree.

Let i be a number assigned to a node in a complete binary tree. Then:

1. If i = 1, then this node is the root of the tree. If i > 1, then the parent of this node is assigned the number (i / 2).

2. If 2*i > n, then the corresponding node has no left child. Otherwise, the left child of that node is assigned the number 2*i.

3. If 2*i + 1 > n, then the corresponding node has no right child. Otherwise, the right child of that node is assigned the number 2*i + 1.

This property suggests a trivial array-based representation of a complete binary

tree, where i is the index of the node in the array. We will see that a slight

modification in this representation allows us to represent any binary tree in a

linear fashion.

The “generic” Binary Tree ADT

We cannot provide a complete specification of the Binary Tree ADT, as we did

with other ADT’s so far, because the hierarchical relationship in the binary tree

cannot be uniquely defined. We define here only a set of basic operations on

binary trees, and more specific binary tree ADT’s will be introduced as the

need arrives.

Operations (methods) on binary trees:

empty () Returns true if the binary tree is empty

getRoot () Returns the root node of the tree

leftChild (node) Returns the left child of node.

rightChild (node) Returns the right child of node.

expandExternal(node) Makes node internal by creating its left and right children

removeAboveExternal(node) Removes an external node together with its parent

insert (node) Inserts node in the appropriate position in the tree

delete (node) Deletes node

preOrder() Visit the root, then the left subtree, then the right subtree

postOrder () Visit the left subtree, then the right subtree, then the root

inOrder() Visit the left subtree, then the root, then the right subtree

levelOrder () Starting from the root, visit tree nodes level by level

Linear (or sequence-based) representation of a binary tree

Linear representation of a binary tree utilizes one-dimensional array of size

2h+1 - 1. Consider the following tree:

+ level 0 (d = 0)

- * level 1 (d = 1)

A B C / level 2 (d = 2)

E F level 3 (d = 3)

To represent this tree, we need an array of size 23+1 - 1 = 15

The tree is represented as follows:

1. The root is stored in BinaryTree[1].

2. For node BinaryTree[n], the left child is stored in BinaryTree[2*n], and the right child is stored in BinaryTree[2*n+1]

i: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

BinaryTree[i]: + - * A B C / E F

Linear representation of a binary tree (cont.)

Advantages of linear representation:

1. Simplicity.

2. Given the location of the child (say, k), the location of the parent is easy to determine (k / 2).

Disadvantages of linear representation:

1. Additions and deletions of nodes are inefficient, because of the data movements in the array.

2. Space is wasted if the binary tree is not complete. That is, the linear representation is useful if the number of missing nodes is small.

Note that linear representation of a binary tree can be implemented by means

of a linked list instead of an array. For example, we can use the Positional

Sequence ADT to implement a binary tree in a linear fashion. This way the

above mentioned disadvantages of the linear representation will be resolved.

Linked representation of a binary tree

Linked representation uses explicit links to connect the nodes. Example:

1

2 5

3 4 6 7

8 9

Nodes in this tree can be viewed as positions in a sequence (numbered 1

through 9).

+

BA

- *

C /

E F

Binary tree nodes (linked representation)class BTNode {

char data;

BTNode leftChild;

BTNode rightChild;

BTNode parent;

int pos;

public BTNode () {

}

public BTNode (char newData) {

data = newData;

}

public BTNode (char newData, BTNode newLeftChild, BTNode newRightChild) {

data = newData;

leftChild = newLeftChild;

rightChild = newRightChild;

}

... methods setData, setLeftChild, setRightChild, getData, getLeftChild, getRightChild,

displayBTNode follow next ...

}

Binary tree (linked representation)

We can use a positional sequence ADT to implement a binary tree. Our

example tree, in this case, we be represented as follows:

position 1 2 3 4 5 6 7 8 9

data + - A B * C / E F

leftChild 2 3 null null 6 null 8 null null

rightChild 5 4 null null 7 null 9 null null

parent null 1 2 2 1 5 5 7 7

class BTLRPS implements PSDLL {

private BTNode header;

private BTNode trailer;

private int size;

int position;

... class methods follow ... }

Traversals of a binary tree

Preorder traversal

public void preOrder (BTNode localRoot) {

if (localRoot != null) {

localRoot.displayBTNode();

preOrder(localRoot.leftChild);

preOrder(localRoot.rightChild); } }

Example Consider a tree with an ordering property, where nodes are inserted in the following order b i n a r y t r e e, i.e.

b

a i

e n

e r

y

t

r

The preorder traversal is: b a i e e n r y t r

Traversals of a binary tree (cont.)

Post-order traversal

public void postOrder (BTNode localRoot) {

if (localRoot != null) {

postOrder(localRoot.leftChild);

postOrder(localRoot.rightChild);

localRoot.displayBTNode(); } }

The nodes in the example tree are traversed in post-order as follows:

a e e r t y r n i b

In-order traversal

public void inOrder (BTNode localRoot) {

if (localRoot != null) {

inOrder(localRoot.leftChild);

localRoot.displayBTNode();

inOrder(localRoot.rightChild); } }

The nodes in the example tree are traversed in in-order as follows:

a b e e i n r r t y

Traversals of a binary tree (cont.)

Level-order traversal

public void levelOrder (BTNode localRoot) {

BTNode[] queue = new BTNode[20];

int front = 0;

int rear = -1;

while (localRoot != null) {

localRoot.displayBTNode();

if (localRoot.leftChild != null) {

rear++;

queue[rear] = localRoot.leftChild; }

if (localRoot.rightChild != null) {

rear++;

queue[rear] = localRoot.rightChild; }

localRoot = queue[front];

front++; } }

The nodes in the example tree are traversed in level-order as follows:

b a i e n e r y t r

Example applications of binary tree traversals

3. Application of an in-order traversal: binary search trees

A binary search tree is a tree with an ordering relationship between data in the

nodes (i.e. all nodes with smaller data are in the left subtree and all nodes with

greater or equal data are in the right subtree). See slide 7 for an example

In-order traversal of a binary search tree produces an ordered list:

32 68 70 74 80 84 86 87 88 90 97 103 109

Binary search trees allow for a very efficient search (in log N time). The idea

of the binary tree search is the following: to find a node with a given datum

(the target), compare the target to the root; if it is smaller, go to the left subtree;

if it is larger, go to the right subtree; if it is equal, stop.

Insertion in binary search tree Insert 9 in the the following tree:

3

2 15

1 11

7 13

Step 1: search for 9 3

2 15

1 11

search stops here 7 13

Step 2: insert 9 at the point where the search terminates unsuccessfully

3

2 15

1 11

7 13 That is, new nodes are always inserted at the leaf level.

9

Binary Tree with an ordering property: the insert method

class BTLRADT {

BTNode root;

public BTLRADT () { }

public BTNode getRoot () {

return root; }

public void insert (char newData) {

BTNode newNode = new BTNode ();

newNode.data = newData;

if (root == null)

root = newNode;

else {

BTNode temp = root;

BTNode parent;

while (true) {

parent = temp;

if (newData < temp.data) { //go left

temp = temp.leftChild;

if (temp == null) {

parent.leftChild = newNode;

return; }

}

else { // go right

temp = temp.rightChild;

if (temp == null) {

parent.rightChild = newNode;

return;

}

}

}

}

}

Deletion in binary search tree Consider the tree:

3 7

2 15 Deleting 3 2 15

1 11 1 11

7 13 13

The following cases of deletions are possible:

1. Delete a note with no children, for example 1. This only requires the appropriate link in the parent node to be made null.

2. Delete a node which has only one child, for example 15. In this case, we must set the corresponding child link of the parent’s parent to point to the only child of the node being deleted.

3. Delete a node with two children, for example 3. The delete method is based on the following consideration: in-order traversal of the resulting tree (after delete operation) must yield an ordered list. To ensure this, the following steps are carried out:

Step 1: Replace 3 with the node with the next largest datum, i.e. 7.

Step 2: Make the left link of 11 point to the right child of 7 (which is null here).

Step 3: Copy the links from the node containing 3 to the node containing 7, and make the parent node of 3 point to 7.

The Tree ADT

Assuming that a general tree is implemented as a positional container, the

following is an incomplete set of methods supported by the data structure:

Container (positional sequence) methods: – empty(): returns true if the container is empty.– node(position): returns the node in position.– elements(): returns an enumeration of all data stored at nodes of the tree.– positions(): returns an enumeration of all the positions (nodes) of the tree.– size(): returns the size of the container.– replace (position, item): replaces the data at position with item.– swap (position1, position2): swaps data in position1 and position2.

Tree specific methods:– getRoot(): returns the root node of the tree– isRoot(position): returns true if the node in position is the root note.– isInternal(position): returns true if the node in that position is an internal node.– isExternal(position): returns true if the node in that position is an external node.– parent(position): returns the parent of the node in position.– children(position): returns a set of children of the node in position.– siblings(position): returns a set of siblings of the node in position.

Computing a node’s depth and a tree’s height The depth of a tree node is a number of ancestors of that node, excluding the node itself. That

is, the depth of the root is 0, while the depth of any other node is the depth of its parent plus

one. The method, depth, can be implemented recursively as follows:

public int depth (int position) {

if (isRoot(position))

return 0;

else

return (1 + depth(parent(position))); }

The height of the tree is equal to the maximum depth of external nodes of the tree. The

method height can be implemented as follows:

public int height () {

int h = 0;

Enumeration nodes = positions();

while (nodes.hasMoreElements()) {

int nextNode = nodes.nextElement ();

if (isExternal(nextNode))

h = Math.max(h, depth(nextNode));

}

return h; }

Binary tree representation of a general tree

Consider the following genealogical tree

Jim

Bill Katy Mike Tom

Dave Mary Leo Bety Rog

Lary Paul Peny Don

We can represent it in the following binary tree format:

1 Jim

2 Bill 8 Katy 10 Mike 14 Tom

3 Dave 4 Mary 9 Leo 11 Bety 13 Rog

5 Lary 6 Paul 7 Peny 12 Don

Binary tree representation of a general tree (contd.)

In the resulting binary tree, the left-child pointer (we can call it here the children

pointer) points to the first child of the ordered list of children, while the right-child

pointer (we can call it here the sibling pointer) points to the next sibling of a node.

We can represent the resulting binary tree as a positional sequence:

position 1 2 3 4 5 6 7 8 9 10 11 12 13 14

data Jim Bill Dave Mery Lary Paul Peny Katy Leo Mike Bety Don Rog Tom

firstChild 2 3 null 5 null null null 9 null 11 12 null null null

sibling null 8 4 null 6 7 null 10 null 14 13 null null null

parent null 1 2 3 4 5 6 2 8 8 10 11 11 10

class TNode {

private String data;

private TNode children, sibling;

int position;

... class methods follow ...

}

Tree traversals

Consider our example tree

Jim

Bill Katy Mike Tom

Dave Mary Leo Bety Rog

Lary Paul Peny Don

Preorder traversal is:

Jim Bill Dave Mary Lary Paul Peny Katy Leo Mike Bety

Don Rog Tom

Postorder traversal is:

Dave Lary Paul Peny Mary Bill Leo Katy Don Bety Rog

Mike Tom Jim

Preorder traversal of a general tree

Preorder traversal works as follows: 1.) select a node and visit it and its children;

2.) go to the next node at the same level and do the same until all of the tree

nodes are processed.

Algorithm preOrder (TNode)

visit TNode

for each child TNodeChild of TNode do

recursively perform preOrder(TNodeChild)

Or, in JAVA:

public void preOrder (TNode localRoot) {

localRoot.displayTNode();

Enumeration localRootChildren = localRoot.children(localRoot.getPosition());

while (localRootChildren.hasMoreElements()) {

TNode nextNode = localRootChildren.nextElement();

preOrder (nextNode); } }

Note: If a general tree is represented as a binary tree, a preorder traversal of the

general tree and the corresponding binary tree, produces the same result.

Postorder traversal of a general treeIn postorder traversal, the tree is processed from left to right, ensuring that no

node is processed until all nodes below it are processed. That is,

Algorithm postOrder (TNode)

for each child TNodeChild of TNode do

recursively perform postOrder(TNodeChild)

visit TNode

Or, in JAVA:

public void postOrder (TNode localRoot) {

Enumeration localRootChildren = localRoot.children(localRoot.getPosition());

while (localRootChildren.hasMoreElements()) {

TNode nextNode = localRootChildren.nextElement();

postOrder (nextNode); }

localRoot.displayTNode(); }

Note: If a general tree is represented as a binary tree, a postorder traversal of the

general tree and the corresponding binary tree, do not generate the same result.

However, the inorder traversal of the corresponding binary tree generates the

same result as the postorder traversal of the general tree.

Ternary tree representation of a general tree

If node siblings in a general tree form ordered lists, then we can represent the

tree as a ternary tree. In a ternary tree, each node has the following attributes:

– left sibling, which is either null or points to a node whose data precedes that of a given node at the same level;

– data stored in the node;

– children, a pointer to the ordered list of children of that node, or null if the node has no children;

– right sibling, which is either null or points to a node whose data equals or follow that of a given node at the same level.

Ternary tree representation of a general tree (contd.)

Consider the example tree:

Jim

Bill Katy Mike Tom

Dave Mary Leo Bety Rog

Lary Paul Peny Don

Represented as a ternary tree, it looks like as follows:

Jim

Bill Katy Mike Tom

Dave Mary Leo Bety Rog

Lary Paul Peny Don

Preorder traversal of a ternary tree

The idea is the following: for each node do 1.) process the node, and 2.) access

the binary tree representing its children, and process this binary tree inorder.

Algorithm preorderTernary (ternaryNode)

if (ternaryNode is internal node) {

preorder (ternaryNode.leftSibling)

process ternaryNode

inorder (ternaryNode.children)

preorder (ternaryNode.rightSibling)

}