Download Lecture 15 Trees

Data Structures
TREES
Azhar Maqsood
School of Electrical Engineering and Computer Sciences
(SEECS-NUST)
What is a tree?
• In computer science, a tree is an abstract
model of a hierarchical structure
• A tree consists of nodes with a parentchild relation
• Previous data structures placed data in
linear order.
• Some data organizations require
categorizing data into groups, subgroups
Tree Terminology
• Trees are a hierarchical data structure of
nodes
• Nodes are linked by edges
4
nodes
2
5
edge
1
3
Applications
• Organization charts
• File systems
• Programming environments
• A class hierarchy in programming languages that
support single inheritance (e.g. Java)
• Document Object Model (for HTML and XML
• Artificial Intelligence (Decision making etc)
tree is hierarchical classification
• A tree is hierarchical classification.
• Data items appear at various levels within the
organization
• E.g., directory structure:
tree is hierarchical classification
Decision Trees
Decision tree based upon expert system
Tree Terminology (Parent / Child)
• Root: node without parent
• A parent node references one or
more nodes (children nodes)
that are “lower” in the tree
hierarchy
• If node u is the parent of node v,
then v is the child of u
• Except for the root (no parent),
every node has exactly one
parent (by definition)
• A tree has only one root node
u
v
root
Tree Terminology (Siblings)
• Two nodes that are children of the same
parent.
Tree Terminology (Internal node)
• A node is internal if it has one or more
children
Tree Terminology (Leaf /External node)
• A node is a leaf if it has no children
Tree Terminology (Ancestor / Descendent)
• An ancestor of a node is either the node’s
parent or any ancestor of the node’s parent
(this is recursive)
• The root is an ancestor of each other node
• Descendant of a node: child, grandchild,
grand-grandchild
u
u
v
v
Tree Terminology (Subtree)
• A tree may be divided into subtrees.
• A subtree is a tree that has the child of a node as its
root.
• Hence, a subtree is composed of a node and all of
that node’s descendants.
• The first node in a subtree is known as the root of
the subtree and is used to name the subtree.
• Subtrees themselves can be further divided into
other subtrees.
Tree Terminology (Subtree)
• The subtree of T rooted at node v is the tree
consisting of all the descendents of v in T
(including v)
• tree consisting of a node and its descendants
Root of subtree
v
v
Tree Terminology (Depth of a node)
• The depth of a node v in T is the number of
ancestors of v, excluding v itself.
• More formally:
• If v is the root, the depth of v is 0
v
depth of v = 1
v
depth of v = 3
Tree Terminology( Height /depth of a tree)
• The depth of a tree is the maximum depth of
any of its leaves
• maximum levels of a tree
tree depth = 3
tree depth = 2
tree depth = 0
Terminology …
Height of the tree?
Depth of node B?
Terminology
Parents: A, B, F
Children: B, E, F, C, D, G, H, I
Siblings: {B, E, F}, {C, D}, {G, H, I}
Leaves: C, D, E, G, H, I
Internal Nodes: A, B, F
Terminology
• Two nodes are adjacent if a branch connects
them.
• A path is a sequence of nodes in which each
node is adjacent to the next one.
• Every node in the tree can be reached by
following a unique path starting from the
root.
• The length of this path is the number of
edges on the path.
• There is a path of length 0 from every node
to itself.
Terminology
• The path from the root, A, to the leaf, I, is denoted
as AFI and has a length of 2.
• ABD is the path from the root, A, to the leaf, D,
and also has a length of 2.
Types of Trees
• General tree – a node
can have any number
of children
• Binary tree – a node
can have at most two
children
Binary Tree
Binary Trees
• The simplest form of tree is a Binary Tree
• A Binary Tree consists of
• (a) A node (called the root node) and
• (b) Left and right subtrees
• Both the subtrees are themselves binary trees
• Note: this is a recursive definition
• (A node can’t have more than 2 children)
Binary tree
General tree
Binary Trees
• Finite set of nodes that is either empty, or
consists of a root and two disjoint binary
trees called the left subtree and right
subtree.
• Node contains information and two pointers to
other nodes
• Each node has at most two children
• A binary tree is either empty or has the
following form:
root
TL
TR
• Where Tleft and Tright are binary trees.
Binary Trees
• Full binary tree: All leaves on the same level
and every node has either zero or two
children.
• Complete binary tree: Leaves are filled from
left to right on one level before moving to
next level.
Binary Trees
Binary Trees
• Skewed binary tree: Contains only left or
right children.
• Similar: Two trees with same structure and
different data.
• Copy or identical: Same structure and
same data.
Tree Height and Full Binary Tree
• If
h = height of a binary tree,
max # of leaves = 2h
max # of nodes = 2h + 1 - 1
• A binary tree with height h
and 2h + 1 - 1 nodes (or 2h
leaves) is called a full binary
tree
Binary tree
Visiting and Traversing a Node
• Many applications require that all of the
nodes of a tree be “visited”.
• Visiting a node may mean printing contents,
retrieving information, making a calculation,
etc.
• Traverse: To visit all the nodes in a tree in a
systematic fashion.
• A traversal can pass through a node without
visiting it at that moment.
Binary Tree Structure
• The representation of a binary tree structure is
relatively straightforward.
• We need a variable to store the data at the
node and 2 pointers to the left and right
subtrees.
struct Node {
int data
Node *left
Node *right
}
Binary Tree Structure