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BEFORE TREES – A more general structure |
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Graph G = (V, E) |
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V = Vertices |
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E = Edges |
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Undirected Graph |
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V = {1,2,3} E = {(1,2),(2,3),(3,1)} |
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Note:
(1,2) and (2,1) are the same edge |
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VERTEX SYNONOMOUS WITH NODE |
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Directed Graph |
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PATHS:
e.g. {(3,3),(3,4),(4,1)} |
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CYCLE: e.g. {(1,2),(2,5),(5,4),(4,1)} |
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ACYCLIC: DIRECTED GRAPH WITH NO CYCLE |
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INDEGREE: NUMBER OF EDGES |
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DIRECTED INTO A VERTEX |
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OUTDEGREE: NUMBER OF EDGES |
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DIRECTED OUT OF A VERTEX |
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DEF: A TREE IS AN ACYCLIC |
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DIGRAPH WHICH HAS ONE |
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NODE OF INDEGREE 0 (ROOT) |
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AND ALL OTHER NODES |
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ARE OF INDEGREE 1 |
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DEF: A FOREST IS A (DISJOINT) UNION |
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OF A NUMBER OF TREES |
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A IS THE ROOT IN ALL 3 EXAMPLES |
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LEVEL OR DEPTH: |
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LEVEL(ROOT) = 0 |
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LEVEL(NODE) = LENGTH OF PATH FROM ROOT TO
NODE (NUMBER OF
EDGES) |
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E.g.
LEVEL(B) = 1, LEVEL(D) = 2 IN
ABOVE |
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NOTE: EVERY NODE (EXCEPT A) IS A |
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DECENDANT OF A |
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B, C, D ARE CALLED SIBLINGS |
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C IS A PARENT OF F |
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HEIGHT OF A VERTEX: |
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THE LENGTH OF THE LONGEST PATH TO A LEAF |
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E.g. HEIGHT(C) = 2 |
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HEIGHT(TREE) = HEIGHT(ROOT) = 3 |
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HEIGHT(E) = 0 |
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TREES, CONT’D |
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DEF: A BINARY TREE IS ONE FOR WHICH |
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1) EACH CHILD IS EITHER A LEFT OR RIGHT
CHILD |
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2) NO VERTEX HAS MORE THAN 2 CHILDREN |
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i.e. Vertex V đ Outdegree(V) ≤
2 |
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E.g.’s: |
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PREORDER (NLR – Node, Left, Right) |
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1) PROCESS ROOT NODE |
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2) TRAVERSE LEFT SUBTREE IN PREORDER |
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3) TRAVERSE RIGHT SUBTREE IN PREORDER |
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POSTORDER (LRN) |
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1) TRAVERSE THE LEFT SUBTREE IN POSTORDER |
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2) TRAVERSE THE RIGHT SUBTREE IN
POSTORDER |
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3) PROCESS THE ROOT NODE |
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E.g. |
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INORDER (LNR) |
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1) TRAVERSE THE LEFT SUBTREE IN INORDER |
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2) PROCESS THE ROOT NODE |
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3) TRAVERSE THE RIGHT SUBTREE IN INORDER |
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E.g. |
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Example Tree Object Methods |
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*Construct (initialize) a tree object |
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*Build a tree object from a given set of
data |
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*Destroy (delete) a tree object |
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*Create a node object in a tree object |
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*Delete a node object in a tree object |
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*Check to see if a tree object is empty |
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*Add/insert a node object specified by a key
value |
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*Search for a node object specified by a key
value |
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*Search for a predecessor (successor) of a
single key element |
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Example Tree Object Methods – Cont’d |
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* Get a specified attribute using a single
search key |
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*Update a node object with new data |
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*Retrieve data contained in a node object |
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*Sort a tree object |
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*Display (print) a tree or subtree object |
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*Print a node object |
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*Traverse a tree object |
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// bc_tree.h : Define an abstract base class
“Tree” |
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Class Tree { |
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public: |
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virtual BOOLEAN is_empty (void) = 0; |
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virtual void build_tree (DATA_TYPE A[]) = 0; |
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virtual void add_node (DATA_TYPE node_data)
= 0; |
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virtual void search_node (DATA_TYPE
srch_key) = 0; |
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virtual void delete_node (DATA_TYPE
srch_key) = 0; |
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virtual void postorder (void) = 0; |
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virtual void preorder (void) = 0 |
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virtual void inorder (void) = 0; |
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Definition: A binary search tree (BST) is a
binary tree; it is either empty or contains a root node along with left and
right binary search subtrees such that: |
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(1) the root node (data object) contains
an attribute designated as the key |
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(2) all nodes in the left subtree of the
root contain keys of value (numerically or alphabetically) less than or
equal to the key of the root node |
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(3) all nodes in the right subtree of
the root contain keys of value greater than the key of the root node |
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Notes