Unformatted text preview: proof, you may make use of any of the equivalences listed on page 30 of the CSC 165 lecture notes —
see the course website for a link to the CSC 165 notes online. Dept. of Computer Science, University of Toronto, St. George Campus Page 2 of 2 Solution
It is important to note that in structural induction, we do induction on the structure of a recursively deﬁned
set of objects. In this case, these are ternary trees. In the base case, we argue that the smallest objects that
belong to this set (that are not recursively deﬁned) satisfy the property. In the induction step, we assume
that if the “ingredients” satisfy the property then, using the rules of the construction for the set, all the
newly constructed elements will also satisfy the property. So, with structural induction, we are not doing
induction on any natural number (such as is the case with simple and complete induction).
Let’s see this proof now.
Basis: The most basic coloured ternary tree is a single node. The single node by deﬁnition is both a binary
tree and monochromatic. So, it does satisfy the property.
Induction Step: Any ternary tree T has to be constructed by taking th...
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 Winter '09
 Logic, Mathematical Induction, Recursion, complete ternary tree

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