CMPSCI 187:Programming With Data Structures
Lecture #26: Trees: Definition and Implementation
16 November 2011
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Trees: Definition and Implementation
• Tree Vocabulary
• Basic Strategies to Implement Trees
• Ways to Traverse a Tree
• Implementing Trees with Links
• Implementing Trees with Arrays
Tree Vocabulary
• Trees are made up of
nodes
.
The key property of a tree is that every node
except for one, the
root
, has exactly one
parent
.
We use genealogical
language throughout, so that children of the same node are
siblings
, a node
that can be reached by following parent pointers is an
ancestor
, and a node
whose ancestor you are is your
descendent
.
• Nodes with no children are called
leaves
, and nodes with children are
internal nodes
.
The root is at
level 0
, and the
level
of any other node is the
number of parent pointers you must follow to reach the root.
The
height
of a
tree is the largest level number of any node in it.
• A tree is
binary
if every internal node has either one or two children, and it is
k-ary
if every node has at most k children.
L&C often implicitly assume that
every internal node has
exactly
two children, but this isn’t a math course...
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Kinds of Binary Trees
• A binary tree is called
balanced
in L&C if its leaves are all on one level or on
two adjacent levels.
A balanced tree of height h has at most 2^h leaves, and
the fewest it can have (with no unary nodes) is 2^{h-1} + 1, still O(2^h).
The
height of a balanced tree with n leaves, or with n nodes, is O(log n).
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- Fall '11
- barrington
- Data Structures
-
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