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Mathematical Binary Tree Notation
24 February 2013 OSU CSE 1
Binary Tree Theory
• Another mathematical model that we will use is that of binary trees
• A binary tree can be thought of as a structure comprising zero or more nodes, each with a label of some mathematical type, say T, called the label type
• We call this math type binary tree of T
24 February 2013 OSU CSE 2
Recursive Structure • A binary tree (type binary tree of T) is
either – the empty tree (empty_tree), which has no nodes
at all; or – a non-empty tree, which consists of:
• A root node (type T) • Two subtrees of the root, each of which is a binary tree (type binary tree of T)
• Since a non-empty binary tree contains two other binary trees (each of which may contain others), its structure is recursive
24 February 2013 OSU CSE 3
Recursive Structure • A binary tree (type binary tree of T) is
either – the empty tree (empty_tree), which has no nodes
at all; or – a non-empty tree, which consists of:
• A root node (type T) • Two subtrees of the root, each of which is a binary tree (type binary tree of T)
• Since a non-empty binary tree contains two other binary trees (each of which may contain others), its structure is recursive
24 February 2013 OSU CSE 4
Previously, with general trees and XMLTree, we downplayed the recursive structure. Not here!
The Empty Tree
• The empty binary tree is denoted by empty_tree and is shown in our diagrams as simply the absence of a tree where there could be one, or (when it’s not completely clear from context) as
24 February 2013 OSU CSE 5
Non-Empty Trees
• Every non-empty tree is the result of the mathematical function compose applied to a value of the label type T and two other values of type binary tree of T, which are the root and the left and right subtrees of the resulting binary tree
• In the diagrams that follow, T is integer
24 February 2013 OSU CSE 6
The compose Function
24 February 2013 OSU CSE 7
x, t1, t2 compose(x, t1, t2)
x = 42
t1 =
t2 =
42
The compose Function
24 February 2013 OSU CSE 8
x, t1, t2 compose(x, t1, t2)
x = 42
t1 =
t2 =
42
Diagrams use color triangles to stand for any non-empty tree.
The compose Function
24 February 2013 OSU CSE 9
x, t1, t2 compose(x, t1, t2)
x = 42
t1 =
t2 =
42
root
left subtree
right subtree
The compose Function
24 February 2013 OSU CSE 10
x, t1, t2 compose(x, t1, t2)
x = 42
t1 =
t2 =
42
The compose Function
24 February 2013 OSU CSE 11
x, t1, t2 compose(x, t1, t2)
x = 42
t1 =
t2 =
42
The compose Function
24 February 2013 OSU CSE 12
x, t1, t2 compose(x, t1, t2)
x = 42
t1 =
t2 =
42
Size, Height, and Labels • The size of t, i.e., the total number of
nodes in t, is denoted by |t| • The height of t, i.e., the number of nodes
on the longest path from the root to an empty subtree of t, is denoted by ht(t)
• The labels of t, i.e., the set whose elements are exactly the labels of t (i.e., the tree’s labels without duplicates and ignoring order) is denoted by labels(t)
24 February 2013 OSU CSE 13
Size
24 February 2013 OSU CSE 14
t |t|
0
1 + |ls| + |rs|
13
ls rs
Height
24 February 2013 OSU CSE 15
t ht(t)
0
1 + max(ht(ls), ht(rs))
13
ls rs
Labels
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t labels(t)
{ }
{13} union labels(ls) union
labels(rs)
13
ls rs
What Is This Tree’s Mathematical Value?
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1 3
2
4
5
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1 3
2
4
5
Note: Its size is 5.
Its height is 3.
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1 3
2
4
5
compose(4, , )
24 February 2013 OSU CSE 20
1 3
2
4
5
compose(2, , )
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1 3
2
4
5
compose(1, , )
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1 3
2
4
5
compose(3, , )
24 February 2013 OSU CSE 23
1 3
2
4
5
compose(5, , )
Its Mathematical Value Must Be...
compose(4, compose(2, compose(1, empty_tree, empty_tree), compose(3, empty_tree, empty_tree)), compose(5, empty_tree, empty_tree)))
24 February 2013 OSU CSE 24
1 3
2
4
5
