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FUNDAMENTALS: DATA STRUCTURES & ALGORITHMS
SOME DEFINITIONS OF GRAPHS AND TREES
GRAPH
an ordered pair G = < V, E > where V is a finite non-empty set of
vertices or nodes and E is a set of edges (unordered pair of vertices).
The Order of the graph is the number of vertices; two distinct vertices
are adjacent is joined by an edge; the degree of any vertex = number
of edges incident on/away from it.
DIRECTED
GRAPH
also known as a DIGRAPH or ORIENTED GRAPH where E defined above - the set of edges - is a set of ordered pairs
from V (subset of V x V)
PATH
the sequence of edges (where consecutive edges share the same vertex)
or vertices; path length is number of edges.
SIMPLE PATH
all vertices are distinct.
CYCLE
a path of length 3 or more which connects v0 with itself; simple cycle
is a cycle whose path is simple.
CONNECTED
GRAPH
every pair of vertices connected by at least one path.
FREE TREE
finite connected graph with no simple cycles.
ORIENTED
TREE
some particular vertex designated as the root;
remaining nodes, if any, partitioned into 1 or more disjoint sets, each
being an oriented tree.
LEVEL OF A
VERTEX
in an oriented tree by 'dangling' all n-1 vertices from the root
we get levels defined as 1 plus the path from the root to vertex. We
also get child, ancestor, sibling, internal/external nodes.
ORDERED TREE
finite set of one or more vertices, one designated the root and the
remaining vertices partitioned in n >= 0 disjoint subsets each of which
is an ordered tree.
BINARY TREE
finite set of vertices that is either empty or consists of a root and two
binary subtrees which are disjoint and are called the left and right
subtrees.
FULL BINARY
TREE
a binary tree in which each node is either a leaf node or has
exactly two non-empty descendants.
COMPLETE
BINARY TREE
a binary tree with leaves on at most two adjacent levels
and those located at the bottom most level lie in the leftmost position
of the tree.
PROPERTY OF
TREES
for a full binary tree:
number of leaf nodes = number of internal nodes + 1