What is a Tree?
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Tree Terminology
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In computer science, a
tree is an abstract model
of a hierarchical
structure
A tree consists of nodes
with a parent-child
relation
Applications:
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– Organization charts
– File systems
– Programming
environments
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Root: node without parent (A)
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Internal node: node with at least
one child (A, B, C, F)
External node (a.k.a. leaf ): node
without children (E, I, J, K, G, H, D)
Ancestors of a node: parent,
grandparent, grand-grandparent,
etc.
Depth of a node: number of
ancestors
Height of a tree: maximum depth
of any node (3)
Descendant of a node: child,
grandchild, grand-grandchild, etc.
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More Tree Terminology
• We use positions to
abstract nodes
• Generic methods:
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integer size()
boolean isEmpty()
objectIterator elements()
positionIterator positions()
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– boolean isInternal(p)
– boolean isExternal(p)
– boolean isRoot(p)
• Update methods:
– swapElements(p, q)
– object replaceElement(p, o)
• Additional update methods
may be defined by data
structures implementing
the Tree ADT
• Java interface: p. 235
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Depth
Basic Algorithms on Trees
• Depth of a node v: number of ancestors of v
• Height of a node v:
if v is a leaf node, height(v) = 0
else height(v) = 1 + maximum height of a child of v
Proposition:
height(T) = maximum depth of a leaf node of T
• Preorder traversal
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• Query methods:
– position root()
– position parent(p)
– positionIterator children(p)
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Postorder
traversal
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• Accessor methods:
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Tree ADT
• Siblings: children of the same parent
• Ordered tree: the children of each node are ordered
• Let n be number of nodes in a tree.
What is the number of edges?
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Subtree: tree consisting of
a node and its
descendants
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• Depth of a node v: number of ancestors of v
• Recursive definition:
if v is the root, depth(v) = 0
else depth(v) = 1 + depth(parent(v))
• Algorithm depth(T, v)
if T.isRoot(v)
return(0)
else return (1 + depth(T, T.parent(v))
• Running time: O(1+dv), dv = depth of node v
assuming isRoot(v) and parent(v) take O(1) time
• Worst case running time: an unbalanced tree
O(n), n = total number of nodes in T
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In practice,
dv << n
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Height
• Recursive definition:
if v is a leaf node, height(v) = 0
else height(v) = 1 + maximum height of a child of v
• Proposition: height(T) = max depth of a leaf node of T
• Algorithm height1(T) {
h=0
for every node v in tree T
if (T.isExternal(v))
h = max(h, depth(T, v));
return h;
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Analysis of height()
We now write an algorithm
using the recursive
definition to see if it’s better.
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Java Code of height1()
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A traversal visits the nodes of a
tree in a systematic manner
In a preorder traversal, a node is
visited before its descendants
Application: print a structured
document
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Algorithm preOrder(v)
visit(v)
for each child w of v
preorder (w)
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An Example
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In a postorder traversal, a
node is visited after its
descendants
Application: compute space
used by files in a directory and
its subdirectories
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Algorithm postOrder(v)
for each child w of v
postOrder (w)
visit(v)
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Postorder Traversal
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Preorder Traversal
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Algorithm height2(T, v) {
if (T.isExternal(v))
return 0;
else {
h = 0;
for every child w of T
h = max(h, height2(T, w));
return (h + 1);
}
}
height2() takes O(n) time
Analysis of height1()
• depth(T, v) takes time
O(1+dv), or O(n) in the
worst case
• The number of leaf nodes of
T is bound by n–1, or O(n)
• Running time of height1() is
O(n2) is the worst case
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Applications
Computing Disk Space
• Either preorder traversal or postorder traversal can be
used when the order of computation is not important.
Example: printing the contents of a tree (in any order)
• Preorder traversal is required when we must perform a
computation for each node before performing any
computations for its descendents.
• Postorder traversal is needed when the computation for
a node v requires the computations for v’s children to be
done first.
Example: Given a file system, compute the disk space
used by a directory.
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Example of Preorder Traversal
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Give an O(n)-time algorithm for computing the depths
of all the nodes of a tree T, where n is the number of
nodes of T. Assume that each node v now has an
additional field depth that stores the depth of v.
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Simple solution: Traversing the tree (preorder or
postorder), call method depth(v) for each node visited
Running time = ?
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Algorithm Depths (v, d) {
v.setDepth(d);
for each child w of v
Depths(w, d+1);
}
• Given a tree T, Depths() is called as follows:
Depths(T.root(), 0);
• Running time = ?
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Example of Postorder Traversal
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Give an O(n)-time algorithm for computing the heights
of all the nodes of a tree T, where n is the number of
nodes of T. Assume that each node v now has an
additional field height that stores the height of v.
Simple solution: Traversing the tree (preorder or
postorder), call method height(v) for each node visited.
Running time = ?
Recall the recursive definition of height:
if v is a leaf node, height(v) = 0
else height(v) = 1 + maximum height of a child of v
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Computing Depths of All Nodes
Recall the recursive definition of depth:
if v is the root, depth(v) = 0
else depth(v) = 1 + depth(parent(v))
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Computing Heights of All Nodes
Algorithm Heights(v) {
h = 0;
for each child w of v
h = max (h, Heights(w));
• Given a tree T, Heights() is
called as follows:
Heights(T.root());
• Running time = ?
if (isExternal(v))
v.setHeight(0);
else
v.setHeight(1+h);
}
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