2. What is a Tree?
• A two dimensional linked structure
• Examples:
– Family tree
– Organization of sports tournament
– Organizational chart of large cooperation
3. A Tree
• A tree is a nonempty collection of vertices and edges that
satisfies certain requirements
• A vertex is a simple object (also referred to as a node) that can
have a name and carry other associated information
• An edge is a connection between two vertices
• 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
• The defining property of a tree is that there is exactly one path
between the root and each of the other nodes in the tree
5. Details
• Each node (except the root) has exactly one node above it,
which is called its parent
• The nodes directly below a node are called its children
• In Figure 1, P is the grandchild of R and has three siblings
• Nodes with no children are sometimes called leaves, or
terminal nodes
• Nodes with at least one child are called non terminal nodes
• Sometimes we refer to non terminal nodes as internal nodes
and terminal nodes as external nodes
6. A Sub Tree
• Any node is the root of a sub tree consisting of it and the nodes
below it
• In Figure 1, there are
– seven one-node sub trees
– one three-node sub tree
– one five-node sub tree
– one six-node sub tree
7. A Forest
• A set of trees is called a forest
• For example, if we remove the root and the edges connecting it
from the tree in Figure 1, we are left with a forest consisting of
three trees rooted at A, R, and E
8. An Ordered Tree
• An ordered tree is one in which the order of the children at
every node is specified
• There are many different ways to draw trees that are not
ordered
9. Levels of a Tree
• The level of a node is the number of nodes on the path from
the node to the root (not including itself)
• For example, in Figure 1, R is on level 1 and S is on level 2
• The height of a tree is the maximum level among all nodes in
the tree (or maximum distance to the root from any node)
• The path length of a tree is the sum of the levels of all the
nodes in the tree (or the sum of the lengths of the paths from
each node to the root)
• The tree in Figure 1 is of height 3 and path length 21
• If internal nodes are distinguished from external nodes, we
speak of internal path length and external path length
10. A Multi Way Tree
• If each node must have a specific number of children
appearing in a specific order, then we have a multi way tree
• In such a tree, it is appropriate to define special external nodes
which have no children (and usually no name or other
associated information)
• Then external nodes act as “dummy” nodes for reference by
nodes that do not have the specified number of children
11. A Binary Tree
• A binary tree is an ordered tree consisting of two types of
nodes: external nodes with no children and internal nodes with
exactly two children
• Since the two children of each internal node are ordered, we
refer to the left child and the right child of internal nodes
• Every internal node must have both a left and a right child,
though one or both of them might be an external node
13. Binary Tree Details
• The purpose of the binary tree is to structure the internal
nodes; the external nodes serve only as placeholders
• A binary tree could be “empty”, consisting of no internal
nodes and one external node
