Unformatted text preview: ree ternary trees of the same height,
let us call these T1 , T2 , and T3 , and placing them under a new root note. In the process of colouring T , we
also colour T1 , T2 , and T3 . And, it is obvious that
Induction Hypothesis: Let us assume that coloured ternary trees T1 , T2 , and T3 satisfy the property, meaning
that they all have monochromatic binary subtrees.
Since we have only two colours (by the pigeon hole principle), 2 of these 3 trees have to have to have
the same colour for their monochromatic subtrees. Without loss of generality, let us assume T1 and T2 both
have blue monochromatic binary subtrees (all the other cases will be similar up to changing the colour or
b
b
the tree names), and we call these (respectively) T1 and T2 . Now the binary tree that consists of the root
b
b
of T and T1 as its left subtree and T2 as its right subtree is binary and monochromatic (all coloured blue)
and a subtree of T . Therefore, we have proved that T also has a monochromatic binary subtree. 1...
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 Winter '09
 Logic, Mathematical Induction, Recursion, complete ternary tree

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