Rooted Trees – CMPT 225

Graphs

Graph is a pair G=<V,E>, with:

• V, a set called “vertacies” or “nodes”
• E, a set of pairs from V, i.e. $E \subseteq V\times V$ called edges
• Example: $G=\langle\{1,2,3\},\{(1,2),(1,3)\}\rangle$

Example in table format:

node	connections
1	2,3
2	1
3	1

“G is directed”: edges are ordered pairs (often called arcs)

$\langle\{\{1,2,3\},\{(1,2),(1,3)\}\}\rangle \neq \langle\{\{1,2,3\},\{(2,1),(1,3)\}\}\rangle$

i.e. graph one:

node	connections
1	2,3
2
3

does not equal graph two:

node	connections
1	3
2	1
3

“G is undirected” edges are sets

$\langle\{\{1,2,3\},\{(2,1),(1,3)\}\}\rangle = \langle\{\{1,2,3\},\{(1,2),(3,1)\}\}\rangle$

i.e. graph one:

node	connections
1	3,2
2	1
3	1

is equal to graph two:

node	connections
1	2,3
2	1
3	1

By default, by “graph” we will mean “undirected graph”.

Path

Path of G of length n is a sequence $\langle V_{0},V_{1},V_{2},\dots V_{n}\rangle$ of vertacies s-t. $(V_{0},V_{1}),(V_{1},V_{2}),(V_{2},V_{3})\dots$ are edges of G.

s-t Path in G: path $\langle s,\dots,t\rangle$ in G.

Example of a s-t path of length 6 for G: $\langle s,d,c,e,f,h,t\rangle$

node	connections
a	b,s
b	a,c
c	b,d,e,s
d	c,s
e	c,f,g
f	e,h
g	e,h,t
h	f,g,t
s	a,c,d
t	g,h

Vertex t is reachable from s in G if ther is an s-t path in G. G is connected if for every pair $u,v\in V$, u is reachable from v.

Transcriber’s note: this is much easier to show with graphics and in fact very hard to show easily without graphics. I did my best, but it should be noted that this is very obvious with a visual graph. It becomes two seperate graphs, in essense. So for convenience, I will be splitting this up into two tables, even though technically it’s one “graph” with two sets of connected nodes.

Connected:

node	connections
a	b,s
b	a,c
c	b,d,e,s
d	c,s
e	c,f,g
f	e,h
g	e,h,t
h	f,g,t
s	a,c,d
t	g,h

Not connected:

node	connections
a	b,s
b	a,c
c	b,d,e,s
d	c,s
e	c
s	a,c,d

node	connections
f	h
g	h,t
h	f,g,t
t	g,h

Cycles & Trees

Cycle in G: path $\langle V_{0},\dots V_{n-1},V_{n}\rangle$ in G where $V_{0}=V_{n}$

Simple path: all vertecies are distinct. Simple Cycle: cycle $\langle V_{0},\dots V_{n-1},V_{n}\rangle$ where $\langle V_{0},\dots,V_{n-1}$ is a simple path.

Simple cylcle of length 5: (see “connected” graph for confirmation): $\langle s,d,c,b,a,s\rangle$

Cycle of length 7 + repeats: $\langle s,d,c,b,f,e,c,s\rangle$

node	connections
a	b,s
b	a,c,f
c	b,d,e,f,s
d	c,s
e	c,f,g
f	c,e,h,b
g	e,h,t
h	f,g,t
s	a,c,d
t	g,h

Tree

Tree: a connected, acyclic graph:

Example 1 (not a tree, due to cycle: $\langle s,c,d,s\rangle$):

node	connections
a	b
b	a,c
c	b,d,e,s
d	c,s
e	c,g
f	h
g	e,h,t
h	f,g,t
s	c,d
t	g,h

Example 2 (a tree, no cycle, all connected):

Now using ARIA trees:

• s
  • a
    • b
  • d
  • c
    • e
      • f
        • h
      • g
        • i

Example 3 (not a tree) (transcriber’s note: written as two trees for simplicity’s sake)

• s
  • a
    • b
  • d
  • c
    • e

• f
  • h
    • g
      • i

Fact: every tree with $n$ verticies has $n-1$ edges.

Rooted-tree:

Rooted tree: tree with a distringuished vertex called the root.

Unrooted tree:

• a
  • b
    • c
    • d
  • e
    • f
      • g
      • h
  • i
    • j
      • k
        • l

Tree T with root:

• a (root)
  • b
    • c
    • d
  • e
    • f
      • g
      • h
  • i
    • j
      • k
        • l

An alternate drawing of T, with no semantic change.

Notice: in a tree there is a unique path between any two verticies. So: a unique path from any vertex to the root. Thus: the root indicates a direction on the edges e.g. towards the root.

A graph of the same tree, T, with arrows pointing towards the root. There is no reason for having any semantic markup for this.

(sometimes “away from the root”).

Rooted Tree Terminology

• z (root, anscestor of a)
  • y
    • x (leaf)
    • w (leaf)
  • v (anscestor of a)
    • u (leaf)
  • t (anscestor of a, parent of a)
    • s (sibling of a, leaf)
    • r (sibling of a, leaf)
    • a (descendant of a, anscestor of a)
      • q (child of a, desandant of a)
        • o (decendant of a, leaf)
        • n (decendant of a, leaf)
        • m (decendant of a, leaf)
      • p (child of a, desendant of a, leaf)

• The root has no parent.
• Leaves have no children.
• Internal nodes are the non-leaves (sometimes root ??? [can’t read] too)

Depth & Height

• 4 (depth 0)
  • 1 (depth 1)
    • 0 (depth 2)
    • 0 (depth 2)
  • 3 (depth 1)
    • 2 (depth 2)
      • 1 (depth 3)
        • 0 (depth 4)
      • 0 (depth 3)
    • 0 (depth 2)

• Depth of node v = length of path from v to the root.
• Height of node v = length of longest path from to a decendant of v (e.g. to a leaf)
• Height of Tree T = height of its root = max height of any node in T = max depth of any node in T.

A rooted tree is:

• k-ary if no node has >k children.
• binary if no node has >2 children.
• ordered if the children of every node are ordered.

E.g. A ordered trenary tree:

• no name
  • 1
    • 1
    • 2
  • 2
  • 3
    • 1
    • 2
    • 3

Notice: when we draw a tree, or represent it in a data structure, we order it.

In an ordered binary tree, every child of a node v is either the “left child of v” or the “right child of v”.

Transcriber’s note: nodes’ children are transcribes from left to right. So for a binary tree, the first child is the “left child” and the second child is the “right child”.

Subtree rooted at v

Subtree rooted at v: tree with root v and containing all decendants of v.

In a binary tree:

• “left subtree of v” means the subtree rooted at the left child of v.
• sim. for “right child of v”.

• z
  • v (subtree rooted at v)
    • y (subtree rooted at v, left subtree of v)
      • v (subtree rooted at v, left subtree of v
      • u (subtree rooted at v, left subtree of v
    • x (subtree rooted at v, right subtree of v)
      • w (subtree rooted at v, right subtree of v)
  • a
    • b
    • c

End