Tree - in “math speak”
An ________ graph is a set of
vertices/nodes and a
set of edges, each edge
connects two vertices.
Any undirected graph in which
every two vertices are connected
by exactly one path (sequence of
edges & associated vertices)
is a ______.
Tree - more properties
A tree is a minimally connected graph (i.e., all vertices are connected by paths,
but removing any edge disconnects)
A tree is a connected graph with one less edge than vertex.
A tree is a maximally acyclic graph (i.e., it contains no cycles, but adding
another edge creates one or more cycles)
Changing Tree Geometry
This is a typical tree to a computer scientist.
properties:
• ______
• _______
• _______
Vertices are often called _____.
The extreme “ends” of a tree
A ____is a node at the extreme end of a path from the root
(i.e., there are no edges to allow the path to continue).
Identify the leaves in the following trees.
Trees can be taller/shorter
A _______of a tree is the number of nodes in its longest path
from root to leaf.
Directed Trees
Tree edges are directed from the root outward.
A ______node shares an edge with every ______.
The parent is “closer to” the root.
Def’n: rooted (directed) tree
Every node, except the root, has exactly
one parent.
The tree’s “extended family”
ancestor
If n1 is the parent of n2,
then n1 is n2’s ancestor.
If n1 is the parent of na and na is an ancestor of n2,
then n1 is n2’s ancestor.
descendant
If n1 is the ancestor of n2,
then n2 is n1’s descendant.
level
The level of a node is the number
of edges in its path to the root.
Subtrees
subtree
Any node of a tree, together with all of its descendants
and associated edges, forms a subtree.
Example: the subtree rooted at F
Expression Trees - An Application
An expression tree is a tree in which a node contains
either an operand or operator and each subtree represents
a subexpression of the whole.
Example: (1 + 2) * (3 - 4) / -(5 * 6)
Decision Trees - Another Application
A decision tree maintains yes/no questions in each non-leaf node,
and each left subtree is for “no” while the right subtree is for “yes”.
Example