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Section 16.6 Binary searches
binary search tree
a tree in which every node has degree \(1\) or \(3\text{,}\) except for a single node of degree \(2\)
initial node
the unique node of degree \(2\) in a binary search tree
terminal node
a node of degree \(1\) in a binary search tree
binary search
the construction of a binary search tree through a series of “either-or” decisions
Worked Example 16.6.1 .
Estimate the root of \(f(x) = 4x^3 + 6x^2 + 3x - 1\) that lies in \((0,1)\) to \(2\) decimal places.
Solution .
The Intermediate Value Theorem from first-year calculus says that if
\(f\) is continuous on the closed interval
\([a,b]\) and
\(f(a),f(b)\) are nonzero and opposite signs, then
\(f\) has a root in the open interval
\((a,b)\text{.}\) We have
\(f(0) = -1 \lt 0\) and
\(f(1) = 12 \gt 0\text{,}\) so there is indeed a root in
\((0,1)\text{.}\) The graph in
Figure 16.6.2 was obtained by performing a binary search by splitting into subintervals.
Figure 16.6.2. A binary search tree search for the root of a polynomial. Since \(f(0.225) \gt 0\text{,}\) the root must be in the subinterval \((0.22,0.225)\text{.}\) This tells us to round down to \(0.22\) instead of rounding up to \(0.23\text{,}\) so we conclude that the root is approximately \(0.22\text{.}\)
