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Lemma:
If node v is the parent
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of
node w then rank(v) >
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rank(w)
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Proof:
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A
node becomes parent of
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another
node only via union
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During
union:
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if one
tree’s rank is larger
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then it
becomes the root
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if the
ranks are tied then the
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new
root’s rank will increase
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In
either case the parent
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ends
up with larger rank
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Lemma:
There are at most n/2r
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nodes
of rank r where n is
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the
number of nodes
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Proof:
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When
node v achieved rank
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r, v
had at least 2r nodes in
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its
tree
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Observe
that those nodes
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cannot
be part of any other
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tree
that achieves rank r
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Thus there are at most n/2r
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nodes
that achieve rank r
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