1. Trees Construction Paths Height/Depth Ordered Trees Traversals
Tree Operations
4/8/2019
CS 303 – Trees
Lecture 7

2. ADT Tree - Terminology Tree The null tree, L, is a tree
A single node, n, is a tree
If n is a node and T1,...,Tk are trees then is a tree
n = Root
T1,...,Tk = subtrees
n1,...,nk = children of n
n = parent of n1,...,nk n n1 n2 nk
4/8/2019
CS 303 – Trees
Lecture 7

3. Paths
If n1,n2,...nk is a sequence of nodes in a tree, such that ni is the
parent of ni+1, i <= i < k, then n1,n2,...,nk is a path
from n1 to nk of length k-1
If there exists a path a,...,b then:
a is an ancestor of b
b is a descendant of a
There is a path of length 0 from a to a. a is both an ancestor
and a descendant of a!
If a is an ancestor/descendant of b and a <> b, then a is a
proper ancestor descendant of b.
4/8/2019
CS 303 – Trees
Lecture 7

4. Height/Depth
The depth of a node n is the length of the (unique!) path from the
root to n. So, Depth(root) = 0.
The height of a node n is the length of the longest path from n to a
leaf node. [A leaf node is one with no descendants]. a 1 b
label each node with height and depth c d 2 e f 3 4 5
4/8/2019
CS 303 – Trees
Lecture 7

5. <> = Order Ordered (the default) – children form a list. a a b d
Unordered (if specified) – children form a set a a = c b d b d c
4/8/2019
CS 303 – Trees
Lecture 7

6. TTLO
For siblings a, b we can define “a is To The Left Of b” (a TTLO b)
For non-siblings, a, b:
If a’, b’ are siblings,
and a’ is an ancestor of a,
and b is an ancestor of b,
and a’ TTLO b’
then a TTLO b a b c d e f g h i k j
work out x TTLO y for various x & y
4/8/2019
CS 303 – Trees
Lecture 7

7. Relationships Must have exactly 1 of: a == b
a is a Proper Ancestor of b
a is a Proper Descendant of b
a TTLO b
b TTLO a
Exercise: Prove it!
4/8/2019
CS 303 – Trees
Lecture 7

8. Traversals
Suppose we “walk” the tree and generate a sequence of nodes.
For leaves, we want a TTLO b a ...,a,...,b,...
What about internal nodes (and root)?
Three choices:
Pre-order: n, (t1), (t2), ..., (tk)
In-order: (t1), n, (t2), ..., (tk)
Post-order: (t1), (t2), ..., (tk), n
4/8/2019
CS 303 – Trees
Lecture 7

9. Traversal Example Pre-Order (Prefix): *+ab+ac
In-Order (Infix): (a+b)*(a+c)
Post-Order(Postfix): ab+ac+*
Prefix and Postfix also known as “(reverse) Polish” after
Jan Lukasiewicz (a Polish logician)
If a TTLO b, which one (pre-, in-, post-order) guarantees that
..., a, ..., b, ...? Prove it!
4/8/2019
CS 303 – Trees
Lecture 7

10. Tree Operations (one choice)
p = Parent(T,n)
lc = LeftmostChild(T,n)
rs = RightSibling(T,n)
l = Label(T,n)
T = Create(n, T1, ..., Tk)
r = Root(T)
4/8/2019
CS 303 – Trees
Lecture 7