Introduction to Trees
Trees
Motivation
When at a node in a singly linked list, there is no choice as to which node can be visited next:
when here
we can
only go here
In a doubly linked list we have a choice of 2 nodes:
if here
then here
or here
If decisions or multiple possibilities are inherent in an application, then more than one or two nodes should be able to
Data structures allowing this include trees and graphs:
Tree
Graph
Trees can be regarded as falling between linear lists and graphs in the hierarchy of structural complexity.
Trees
Tree Definitions
There are many ways of defining a tree. We will look at 4 definitions, and as we do, we will define many other aspects
1) Definition in terms of Graphs:
A tree is a connected, undirected graph that contains no circuit.
In other words, there is a unique path between any 2 nodes.
Not a tree, because there is a circuit:
We use the adjective connected since one graph can be considered as consisting of two disconnected parts:
The adjective undirected allows the definition to be more general:
directed graph or digraph:
considered as one graph
undirected graph:
(restricted, since you can’t go against an(general,
arrow) since you can go both ways be
Trees
Because computer applications of trees require that we start at one designated node in the tree in order to reach the
root
Using directed edges allows us to designate the node where we should start as the root node.
Definition: The root node has indegree of 0.
Indegree ≡ The number of directed edges pointing at a node.
Outdegree ≡ The number of directed edges pointing away from a node.
Leaf nodes have outdegree of 0.
For Example:
This root has:
indegree = 0 and
outdegree = 4
This internal node has:
indegree = 1 and
outdegree = 3
Trees
To impose some standardization so that we can visually pick out the root more easily, computer scientists requ
bottom rooted
top rooted
side
rooted
(European)
(American)
(compromise)
When we root a directed tree, it is no longer necessary to draw the arrows on the directed edges. They are und
(From now on, we will usually draw trees as
Trees
2) Definition in terms of Number of Edges and Nodes:
A tree is a finite connected graph such that:
no. of Nodes = no. of Edges + 1
For Example:
3 nodes
2 edges
5 nodes
4 edges
8 nodes
7 edges
3) Definition in terms of Connectivity:
A graph is a tree iff the removal of any edge would disconnect the graph:
For Example:
Definition: A collection of disconnected trees is called a forest:
For Example:
Removing this, or any other edge, disconnects the g
A forest of 5 trees.
Trees
4) Recursive Definition
(Defining trees in terms of trees.)
(This is the definition that computer scientists prefer.)
A tree is a finite set of nodes which is either empty (root=NULL) or there is
one node called the root node and the remaining nodes can be partitioned
into disjoint sets each of which is also a tree.
These 3 disjoint partitions are also trees – and with respect to the larger structure, they are called subtrees.
3
1
2
This recursive definition for trees gives us some easily defined search algorithms, which we will study later.
Trees
Defining Parts of a Tree:
Many scientific and medical terms seem obscure to us because these disciplines originated millennia ago within ancie
Many medical terms derive from ancient Greek.
stetho·scope
Greek: lungs
ophthalmo·scope
eye
to watch
osteo·myel·itis
bone marrow inflamation
Even though the terminology came from common, everyday words, it is now foreign to us because of the language di
More recent scientific disciplines kept Latin as their common language of discourse.
Fortunately for us, computer science did not make the mistake of obscuring its definitions in ancient Latin or Greek.
It uses current, common everyday English words like mouse, bugs, chips, etc. (More obscure terms mainly come fro
However, there is a drawback. For some concepts within our discipline, there may be more than one common word,
Botanical
Generic
predecessor
successor
Familial Relations
ancestor – parent – father →
descendent – child – son →
brothers or siblings
Generic
← root ------------------ start node
← branch ------------- edge or arc
← branch node ---- internal node
← leaf node ---- external node
↑
vertex
is another name for node
Trees
Kinds of Trees – in terms of branching factor (outdegree).
Unary Trees – outdegree for every node ≤ 1. (a singly linked list)
Binary Trees – outdegree for every node ≤ 2.
Ternary Trees – outdegree for every node ≤ 3.
.
.
.
m - ary Trees – outdegree for every node ≤ m.
m
Note: The indegree for every node in a directed tree, except the root, is 1.
Trees
Mathematical Properties of Trees
Max no. of Nodes per Level:
.
.
.
Binary Tree
Ternary Tree
m-ary Tree
0
1
1
1
2
3
2
4
9
3
8
27
m3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
In general:
h
2h
3h
mh



.
.
.
Level
1
.
.
.
m
.
.
.
m2
.
.
.
.
.
.
.
.
.
Total max no. of nodes:
h
2k
k 0
.
.
.
h
3k
k 0
3 h 1  1

2
.
.
.
These formulas may be proved by induction:
 2
h 1
1
h
mk
k 0
m h 1  1

m 1
Trees
Definition
The path length of a node in a tree is the no. of edges from the root to that node.
path length = 0
path length = 1
height
(or depth)
=4
path length = 2
path length = 3
path length = 4
The height or depth of a tree is the max over all the path lengths of its nodes.
Trees
Definition
An m-ary tree is regular if every one of its internal nodes has exactly m sons.
For each of the following kinds of trees, state which are regular and which are not regular:
Binary
Ternary
regular
regular
regular
not regular

not regular
regular

Trees
Definition
Question: For a regular binary tree of height h, what is the maximum no. of leaves it can have?
Answer: 2h
Question: What is the max no. of leaves for a regular m-ary tree of height h ?
Answer: mh
Such trees are called full trees. For example:
Full Binary Trees
non-Full Binary Trees
Trees
Definition
The min no. of leaves that a regular m-ary tree of height h can have is:
1 + h(m-1)
An example, for binary trees:
m = 2
h = 4
So for a binary tree of height 4, the min no. of leaves for such a tree is:
1 + 4(2-1)
=
1 + 4
=
5
An example, for ternary trees:
m = 3
h = 5
So for a ternary tree of height 5, the min no. of leaves for such a tree is:
1 + 5(3-1)
=
1 + 10
=
11
Trees
Definition
Complete Binary Tree – Such a tree of height h is a full tree up to level h-1 and is filled in from left-to-right a
For example:
Complete Binary Trees
.
.
.
Not Complete
Trees
More Mathematical Properties
Question: How is the height h related to the total number of nodes, n, in a:
1) full binary tree?
2) full m-ary tree?
Recall that the no. of nodes for a full binary tree of height h is:
2h+1−1
So, to get h in terms of n just solve the following equation for h:
2h+1−1 = n
2h+1
= n+1
h + 1 = log2 (n + 1)
h = log2 (n + 1) − 1
Similarly, since mh+1−1 = n for full m-ary trees, the solution is easily determined to be:
m −1
h = logm [n(m−1) + 1] − 1
Questions: Will these log functions always return integer values? Why, or why not?
Trees
Mathematical Properties
Question: How do we determine h in terms of n for complete binary trees?
Hint: Derive the formula by looking at the trees below, and
filling in and observing the number pattern in the table:
1 node
h=0
5 nodes
h =2
2 nodes
h=1
6 nodes
h=2
3 nodes
h=1
7 nodes
h=2
height
h
4 nodes
h=2
8 nodes
h=3
nodes
n
0
1
1
2
1
3
2
4
2
5
Note that the height h increases by 1 at every power of 2, and is simply related as the logarithm
of
2
6 that power of 2
2
7
But the log2 of the intervening integers results in decimal numbers, not integers. How can we denote the truncati
3
8
For complete binary trees: h = ⎿log2(n)⏌
⋮
⋮
For complete m-ary trees: h = ⎿logm(n)⏌
The pair of delimiters ⎿ ⏌ is called the floor function, and is defined as:
⎿ x⏌ ≡ the largest integer ≤ x – e.g. ⎿2.135⏌ = 2 ⎿2.975⏌ = 2
The ceiling function ⎾ ⏋ is the complement of the floor function, and is defined as:
⎾ x⏋ ≡ the smallest integer ≥ x– e.g. ⎾2.135⏋ = 3
⎾2.975⏋ = 3